Understanding Vector Formulas in Physics

In today's lesson, we're going to talk about vector formulas, particularly as it relates to physics. So let's say we have this vector, we'll call it vector v. Vector v has an x component, which we can call v sub x, and it has a y component, v sub y. And let's say this is the angle of that right triangle. The x component of vector v is v cosine theta. The y component, v y, is equal to v sine theta. Now sometimes you may see a vector written simply as v, and other times you may see an arrow on top of it. If you see an arrow, that tells you that it's a vector. Now, if you see it within an absolute value sign, this means you're talking about the magnitude of the vector. Remember, a scalar quantity has magnitude only, but no direction. A vector quantity has both magnitude and direction. But seeing this symbol here, this simply tells you that this is just the magnitude of V without the direction. This vector includes both the magnitude and the direction, and we'll talk more about that later. But the magnitude of vector v for a two-dimensional vector is vx squared plus vy squared. So you get this formula by using the Pythagorean theorem, a squared plus b squared equals c squared. When you solve for c, it's the square root of a squared plus b squared. For a three-dimensional vector, let's say when it has three components, x, y, and z, the magnitude of v is going to be vx squared plus vy squared plus vz squared. So that's if you have a 3D vector. Now the angle, the angle theta in this triangle, you need to find the direction of the vector. For a 2D vector, it's arc tangent Vy over Vx. Now vector V. can be written like this. It can be written in terms of its components. vx i plus vy j if you have a 2d vector. For a 3d vector you'll add plus vz k. But let's focus mostly on 2d vectors for the moment. Now vector v can also be written this way. The magnitude of V, which is basically what we have here, you can write it like this, times cosine theta I plus the magnitude of V times sine theta J. So you can also write a vector in terms of its components using the standard unit vectors I, J, and K. A unit vector is a vector whose length or whose magnitude is always 1, and its purpose is to give direction to another vector. But the standard unit vectors i, j, and k, here's what they equal. So the standing unit vector I is 1, 0, 0. It basically has a length of 1 in the x direction. This is x, y, z. unit vector J is 0 1 0 so it has a length of 1 in the Y direction and the standard unit vector K is 0 0 1 it has a length of 1 in the Z direction so if we were to draw a picture my drawing skills are not the best but Let's make it work. Let's say this is the x-axis, this is the y-axis, and this is the z-axis. And let's say this is the length of 1, this is the length of 1, and this is the length of 1. This would be the unit vector i, this will be j, and this is k. So they all have a magnitude of 1 in their corresponding direction. Now let's go over some more formulas. Let's say we have two vectors. We'll call the first vector vector A. So I'm going to put an arrow on top of it. And the second vector, we'll call it vector B. And there's an angle theta between them. Let's say if you want to multiply two vectors. The first method is called the dot product. The dot product, the result will give you a scalar quantity. So you're multiplying two vector quantities, A and B, and you're going to create a scalar quantity, something that's not a vector. The formula for the dot product is you take the magnitude of A, multiply it by the magnitude of B, and multiply it by cosine of the angle between them. And that will give you the dot product. What you're really doing here is you're multiplying the x components of the two vectors, and you're going to add it to the product of the y components of the two vectors. So you're multiplying the respective x components of each vector and the respective y components, and then you're going to add them. This is for a 2d vector that only operates in the x and y direction. For a three-dimensional vector, you would add az times bz. Now, just like the magnitude of V, the magnitude of A is equal to the square root of AX squared plus AY squared plus AZ squared for a 3D vector. 2D vector, just take out the z, and it's just going to be the square root of ax squared plus ay squared. Now, you would find b in a similar fashion. It's the square root of bx squared plus by squared plus bz squared. Now, a good example of the dot product formula is the calculation of work. Work is force times displacement times cosine of the angle. So let's say if you have a force vector and if you have the displacement vector and you know the angle and you want to calculate the work done by this force you would use this equation so work is basically the dot product of the force and the displacement vector when the force and the displacement vector When they're at 90 degree angles, the work done by this force is zero, because cosine 90 is zero. When the force and the displacement vector are parallel to each other, then the work done is going to be the maximum that that force can deliver, because cosine zero is one. So it's just going to be F times D. So when you multiply... two vectors using the dot product formula, you're going to get a scalar quantity. So that's the scalar dot product formula. Now there's another way to multiply two vectors, and it's the cross product. So this is the vector cross product formula. A cross B is equal to the magnitude of A times the magnitude of B, but instead of cosine theta, it's now sine theta. This is equivalent. to find in the determinant of a 3x3 matrix like this. Let's say that you have I, J, and K in the first row, and then AX, AY, AZ, and then BX, BY, BZ. So when you evaluate this 3x3 determinant, it will give you the value of the cross product. So the dot product, when you multiply two vectors, it gave you a scalar quantity. For the cross product form, When you multiply two vectors, it will give you a vector quantity. So the result of this formula is another vector. Now, the determinant of this 3x3 matrix. By the way, for those of you who want to print out of this formula sheet, or the formulas that I'm going over, feel free to check out the links in the description section below. Now, if you want to know how to evaluate the determinant of a 3x3 matrix, I have a video on that on YouTube. If you type in determinant 3x3 matrix organic chemistry tutor in the search bar, you should see it come up. But here's another way in which you can evaluate this determinant for those of you who want to use a formula. I'm going to need a little more space here. So it's equal to the y component of A times the z component of B minus the z component of A times the y component of B. By the way, if you have a two-dimensional vector, z will have a value of zero. So everywhere you see a z, just put a zero for a 2D vector. But this formula is designed for 3D vectors, but it can work for both. So this will give you the x component of the new vector. So let's call the new vector vector c. So when you multiply vector a and b using the cross product formula, it'll give you a new vector, which we can call vector c. So this quantity here will be the x component of vector c, which you can call cx. Now the y component of vector c is this. It's the z component of a times the x component of b times the x component of a times the z component of b. So the fact that you see the unit vector j tells you that this is going to be the y component of the new vector c. And then we're going to take the x component of A, multiply it by the y component of B, minus the y component of A times the x component of B, and this will give us the z component of C. So just to recap, this part will be the x component of C. Let's use a different color. And this part will give you the y component of the new vector, c. And this part will give you the z component. So let's talk about what's happening here. For the scalar dot product formula, notice that we multiplied the parallel components. We multiplied the x component of a and b, and we multiplied the y component of a and b, and then the z components of a and b. So we only multiplied the parallel components together, and that gave us not a vector quantity but a scalar quantity. With the cross product formula, notice we're multiplying different components. When we multiply the x and the y component of a and b, it gives us the z component of the new vector c. When we multiply the y and z components of vectors a and b, it gives us not the y component or the z component, but the other x component of a new vector c. When we multiply the z and x components of vectors a and b, we get the other component, the y component, for the new vector c. So with the cross product formula, you're multiplying different... components to get basically a new component that's perpendicular to the two components that you multiplied. So this component, the x component, is perpendicular both to the y component and the z component. So whenever you use the cross product formula, you get a new vector that's perpendicular to the two vectors that created it. Now I do need to add one small correction to this formula. This part here gives you the magnitude of the new vector C. So in order for this to be a vector quantity, we need to add a unit vector to it. So I'm just going to put U for unit vector. Sometimes you'll see like in physics, a hat, like R hat, that's also a unit vector. So sometimes you may see it like this. Now, I could have wrote that formula this way. So if I put A cross B, but inside of the absolute value symbol, then this tells me this is just the magnitude of the new vector C. which I can write it as the magnitude of A times B times sine theta without the unit vector. But because I don't have the absolute value around it, I need to put the direction for this vector as well. So this part is the magnitude of the new vector C, and the unit vector gives the direction for vector C. We'll talk more about unit vectors soon, because there's another formula you need to be familiar with. Now, a good example of the cross product formula is the calculation of torque. So torque is the cross product of the lever arm and the force. R is the lever arm. Sometimes you may see L in certain physics textbooks. The magnitude of that torque can be calculated by taking the magnitude of r multiplying by the magnitude of f times sine of the angle. So let's say this is the force vector and this is the lever arm. These two are vectors. and theta is the angle between them. So that's an example of using the cross-product formula in physics when you study torques. Another example is the magnetic force on a moving charge. So let's say if you have a charge like a proton, and it has a velocity vector v, and it's moving in the magnetic field B. The magnetic field will exert a magnetic force on its charge. And the formula to calculate that magnetic force... It's the cross product of the velocity vector and the magnetic field times the charge. The charge is a scale of quantity. Charge doesn't have direction, but velocity and magnetic field have direction. The velocity could be an F. direction, it could be in the Y direction, it could be at an angle, and the same is true with the magnetic field. It can go north, south, east, west, up, down. These are vectors with direction. But Q, the charge, is the scale of quantity, so we won't have an error on time. top of it. So that's going to be the vector cross product formula for the magnetic force on a moving charge. If you just want to find the magnitude of this vector, you could use this part of the formula. So it's going to be the velocity vector times the magnetic field times q and then just sine theta. But now if you want the entire vector itself, it's v cross b times q times sine theta and then you need to multiply by the unit vector which is r hat. So notice every vector can be broken down into its magnitude and its direction. Let's talk about unit vectors. If you want to calculate the unit vector of any other vector, it's equal to that vector divided by its magnitude. Now, if you rearrange this equation, if you multiply both sides by the magnitude of v, you get that the vector v is equal to the magnitude times r hat. A lot of formulas in physics with calculus, if you're taking that course, you'll see that We'll define a vector usually in this format. It's magnitude times direction. R hat is basically the unit vector of the position vector, which means we need to talk about position vectors real quick. So let's say this is z, this is x, this is y. And starting from the origin, we have the position vector, r. To define the position vector art, we need two points. The first point ideally would be the origin, so it would be EZ000. The second point B will have the points XB, YB, ZB. A will be XA, YA, ZA. But to determine the value of... Let me color code this. To determine the value of the position vector, it's going to be the difference between xB and xA. This will give you the x component of the position vector, so that's rx. And then this is going to be yb minus ya, which if a is the origin, ya is just 0. And this is times the unit vector j. And then this is going to be zb minus za times the unit vector k. So this is... Rx, i, and then this is Ryj, and this is Rzk. Now, R hat, it's basically the position vector divided by the magnitude of R. So you have the position vector r with this formula, which I'm about to delete. And you know the magnitude of r is going to be the square root of rx squared plus ry squared plus rz squared. So anytime you need to find the position, the unit vector, r hat, you could use this formula. And you have the formula for the position vector r and the magnitude of r. So going back to the previous formula that we had, the magnetic force on a moving charge the cross product vector formula V cross B Times Q times sine theta and then times R hat. A lot of formulas in physics, you'll see R hat is added. Just remember, R hat is just the unit vector of the position vector. So this here represents the magnitude of the magnetic force on the moving charge. And this part here gives you the direction of that vector. Anytime you take the magnitude of a vector and you multiply it by its unit vector, you're going to get the value of that vector with both magnitude and direction included. Another example of this formula is the electric field vector. Every charge emits an electric field. So if you have a positive charge like a proton, it will emit electric fields in all directions emanating from that proton. To calculate the electric field, or the electric field vector, at some point P, here's the formula for it. It's k, which is a constant, times q, the magnitude of this charge, divided by r squared, where r is the distance between the point of interest and the charge, times r hat. So this part right here represents the magnitude of the electric field vector. r hat is the unit vector of the position vector. It gives this vector direction. When you multiply this, you're going to get Ex, Ey, if it's a 2D vector, and then Ez as well, if it's a 3D vector, when you work out this formula. So I actually have a video on YouTube with this exact process. It explains how to use this formula to get the electric field vector in... It's unit vector notation, where you have the standard unit vectors i, j, and k. So if you want to see how to work out an example problem, type in electric field vector, r-hat, organic chemistry tutor in the YouTube search bar, and you're going to see how to use this formula. So next time when you look at your physics textbook and you see this r-hat in the equations, you'll know what to do with it. By the way, for those of you who want the formula sheet with all these vectors, formulas, and more, feel free to check out the links in the description section below. There are more formulas that are associated with vectors. I didn't cover everything in this video. But if you want to see what those other formulas are, feel free to download that formula sheet for those of you who might be interested in it. But I'm going to stop it here today for this video.

Now sometimes you may see a vector written simply as v, and other times you may see an arrow on top of it. If you see an arrow, that tells you that it's a vector. Now, if you see it within an absolute value sign, this means you're talking about the magnitude of the vector. Remember, a scalar quantity has magnitude only, but no direction. A vector quantity has both magnitude and direction.

But seeing this symbol here, this simply tells you that this is just the magnitude of V without the direction. This vector includes both the magnitude and the direction, and we'll talk more about that later. But the magnitude of vector v for a two-dimensional vector is vx squared plus vy squared.

So you get this formula by using the Pythagorean theorem, a squared plus b squared equals c squared. When you solve for c, it's the square root of a squared plus b squared. For a three-dimensional vector, let's say when it has three components, x, y, and z, the magnitude of v is going to be vx squared plus vy squared plus vz squared. So that's if you have a 3D vector. Now the angle, the angle theta in this triangle, you need to find the direction of the vector.

For a 2D vector, it's arc tangent Vy over Vx. Now vector V. can be written like this. It can be written in terms of its components.

vx i plus vy j if you have a 2d vector. For a 3d vector you'll add plus vz k. But let's focus mostly on 2d vectors for the moment. Now vector v can also be written this way.

The magnitude of V, which is basically what we have here, you can write it like this, times cosine theta I plus the magnitude of V times sine theta J. So you can also write a vector in terms of its components using the standard unit vectors I, J, and K. A unit vector is a vector whose length or whose magnitude is always 1, and its purpose is to give direction to another vector. But the standard unit vectors i, j, and k, here's what they equal. So the standing unit vector I is 1, 0, 0. It basically has a length of 1 in the x direction.

This is x, y, z. unit vector J is 0 1 0 so it has a length of 1 in the Y direction and the standard unit vector K is 0 0 1 it has a length of 1 in the Z direction so if we were to draw a picture my drawing skills are not the best but Let's make it work. Let's say this is the x-axis, this is the y-axis, and this is the z-axis. And let's say this is the length of 1, this is the length of 1, and this is the length of 1. This would be the unit vector i, this will be j, and this is k.

So they all have a magnitude of 1 in their corresponding direction. Now let's go over some more formulas. Let's say we have two vectors. We'll call the first vector vector A. So I'm going to put an arrow on top of it.

And the second vector, we'll call it vector B. And there's an angle theta between them. Let's say if you want to multiply two vectors. The first method is called the dot product.

The dot product, the result will give you a scalar quantity. So you're multiplying two vector quantities, A and B, and you're going to create a scalar quantity, something that's not a vector. The formula for the dot product is you take the magnitude of A, multiply it by the magnitude of B, and multiply it by cosine of the angle between them.

And that will give you the dot product. What you're really doing here is you're multiplying the x components of the two vectors, and you're going to add it to the product of the y components of the two vectors. So you're multiplying the respective x components of each vector and the respective y components, and then you're going to add them.

This is for a 2d vector that only operates in the x and y direction. For a three-dimensional vector, you would add az times bz. Now, just like the magnitude of V, the magnitude of A is equal to the square root of AX squared plus AY squared plus AZ squared for a 3D vector. 2D vector, just take out the z, and it's just going to be the square root of ax squared plus ay squared. Now, you would find b in a similar fashion.

It's the square root of bx squared plus by squared plus bz squared. Now, a good example of the dot product formula is the calculation of work. Work is force times displacement times cosine of the angle. So let's say if you have a force vector and if you have the displacement vector and you know the angle and you want to calculate the work done by this force you would use this equation so work is basically the dot product of the force and the displacement vector when the force and the displacement vector When they're at 90 degree angles, the work done by this force is zero, because cosine 90 is zero.

When the force and the displacement vector are parallel to each other, then the work done is going to be the maximum that that force can deliver, because cosine zero is one. So it's just going to be F times D. So when you multiply...

two vectors using the dot product formula, you're going to get a scalar quantity. So that's the scalar dot product formula. Now there's another way to multiply two vectors, and it's the cross product. So this is the vector cross product formula.

A cross B is equal to the magnitude of A times the magnitude of B, but instead of cosine theta, it's now sine theta. This is equivalent. to find in the determinant of a 3x3 matrix like this. Let's say that you have I, J, and K in the first row, and then AX, AY, AZ, and then BX, BY, BZ. So when you evaluate this 3x3 determinant, it will give you the value of the cross product.

So the dot product, when you multiply two vectors, it gave you a scalar quantity. For the cross product form, When you multiply two vectors, it will give you a vector quantity. So the result of this formula is another vector. Now, the determinant of this 3x3 matrix. By the way, for those of you who want to print out of this formula sheet, or the formulas that I'm going over, feel free to check out the links in the description section below.

Now, if you want to know how to evaluate the determinant of a 3x3 matrix, I have a video on that on YouTube. If you type in determinant 3x3 matrix organic chemistry tutor in the search bar, you should see it come up. But here's another way in which you can evaluate this determinant for those of you who want to use a formula. I'm going to need a little more space here.

So it's equal to the y component of A times the z component of B minus the z component of A times the y component of B. By the way, if you have a two-dimensional vector, z will have a value of zero. So everywhere you see a z, just put a zero for a 2D vector. But this formula is designed for 3D vectors, but it can work for both.

So this will give you the x component of the new vector. So let's call the new vector vector c. So when you multiply vector a and b using the cross product formula, it'll give you a new vector, which we can call vector c. So this quantity here will be the x component of vector c, which you can call cx.

Now the y component of vector c is this. It's the z component of a times the x component of b times the x component of a times the z component of b. So the fact that you see the unit vector j tells you that this is going to be the y component of the new vector c. And then we're going to take the x component of A, multiply it by the y component of B, minus the y component of A times the x component of B, and this will give us the z component of C. So just to recap, this part will be the x component of C.

Let's use a different color. And this part will give you the y component of the new vector, c. And this part will give you the z component.

So let's talk about what's happening here. For the scalar dot product formula, notice that we multiplied the parallel components. We multiplied the x component of a and b, and we multiplied the y component of a and b, and then the z components of a and b.

So we only multiplied the parallel components together, and that gave us not a vector quantity but a scalar quantity. With the cross product formula, notice we're multiplying different components. When we multiply the x and the y component of a and b, it gives us the z component of the new vector c.

When we multiply the y and z components of vectors a and b, it gives us not the y component or the z component, but the other x component of a new vector c. When we multiply the z and x components of vectors a and b, we get the other component, the y component, for the new vector c. So with the cross product formula, you're multiplying different...

components to get basically a new component that's perpendicular to the two components that you multiplied. So this component, the x component, is perpendicular both to the y component and the z component. So whenever you use the cross product formula, you get a new vector that's perpendicular to the two vectors that created it.

Now I do need to add one small correction to this formula. This part here gives you the magnitude of the new vector C. So in order for this to be a vector quantity, we need to add a unit vector to it. So I'm just going to put U for unit vector. Sometimes you'll see like in physics, a hat, like R hat, that's also a unit vector.

So sometimes you may see it like this. Now, I could have wrote that formula this way. So if I put A cross B, but inside of the absolute value symbol, then this tells me this is just the magnitude of the new vector C.

which I can write it as the magnitude of A times B times sine theta without the unit vector. But because I don't have the absolute value around it, I need to put the direction for this vector as well. So this part is the magnitude of the new vector C, and the unit vector gives the direction for vector C.

We'll talk more about unit vectors soon, because there's another formula you need to be familiar with. Now, a good example of the cross product formula is the calculation of torque. So torque is the cross product of the lever arm and the force. R is the lever arm. Sometimes you may see L in certain physics textbooks.

The magnitude of that torque can be calculated by taking the magnitude of r multiplying by the magnitude of f times sine of the angle. So let's say this is the force vector and this is the lever arm. These two are vectors. and theta is the angle between them. So that's an example of using the cross-product formula in physics when you study torques.

Another example is the magnetic force on a moving charge. So let's say if you have a charge like a proton, and it has a velocity vector v, and it's moving in the magnetic field B. The magnetic field will exert a magnetic force on its charge.

And the formula to calculate that magnetic force... It's the cross product of the velocity vector and the magnetic field times the charge. The charge is a scale of quantity.

Charge doesn't have direction, but velocity and magnetic field have direction. The velocity could be an F. direction, it could be in the Y direction, it could be at an angle, and the same is true with the magnetic field. It can go north, south, east, west, up, down. These are vectors with direction.

But Q, the charge, is the scale of quantity, so we won't have an error on time. top of it. So that's going to be the vector cross product formula for the magnetic force on a moving charge. If you just want to find the magnitude of this vector, you could use this part of the formula.

So it's going to be the velocity vector times the magnetic field times q and then just sine theta. But now if you want the entire vector itself, it's v cross b times q times sine theta and then you need to multiply by the unit vector which is r hat. So notice every vector can be broken down into its magnitude and its direction. Let's talk about unit vectors.

If you want to calculate the unit vector of any other vector, it's equal to that vector divided by its magnitude. Now, if you rearrange this equation, if you multiply both sides by the magnitude of v, you get that the vector v is equal to the magnitude times r hat. A lot of formulas in physics with calculus, if you're taking that course, you'll see that We'll define a vector usually in this format. It's magnitude times direction. R hat is basically the unit vector of the position vector, which means we need to talk about position vectors real quick.

So let's say this is z, this is x, this is y. And starting from the origin, we have the position vector, r. To define the position vector art, we need two points. The first point ideally would be the origin, so it would be EZ000.

The second point B will have the points XB, YB, ZB. A will be XA, YA, ZA. But to determine the value of... Let me color code this. To determine the value of the position vector, it's going to be the difference between xB and xA.

This will give you the x component of the position vector, so that's rx. And then this is going to be yb minus ya, which if a is the origin, ya is just 0. And this is times the unit vector j. And then this is going to be zb minus za times the unit vector k.

So this is... Rx, i, and then this is Ryj, and this is Rzk. Now, R hat, it's basically the position vector divided by the magnitude of R.

So you have the position vector r with this formula, which I'm about to delete. And you know the magnitude of r is going to be the square root of rx squared plus ry squared plus rz squared. So anytime you need to find the position, the unit vector, r hat, you could use this formula. And you have the formula for the position vector r and the magnitude of r. So going back to the previous formula that we had, the magnetic force on a moving charge the cross product vector formula V cross B Times Q times sine theta and then times R hat.

A lot of formulas in physics, you'll see R hat is added. Just remember, R hat is just the unit vector of the position vector. So this here represents the magnitude of the magnetic force on the moving charge.

And this part here gives you the direction of that vector. Anytime you take the magnitude of a vector and you multiply it by its unit vector, you're going to get the value of that vector with both magnitude and direction included. Another example of this formula is the electric field vector. Every charge emits an electric field. So if you have a positive charge like a proton, it will emit electric fields in all directions emanating from that proton.

To calculate the electric field, or the electric field vector, at some point P, here's the formula for it. It's k, which is a constant, times q, the magnitude of this charge, divided by r squared, where r is the distance between the point of interest and the charge, times r hat. So this part right here represents the magnitude of the electric field vector. r hat is the unit vector of the position vector. It gives this vector direction.

When you multiply this, you're going to get Ex, Ey, if it's a 2D vector, and then Ez as well, if it's a 3D vector, when you work out this formula. So I actually have a video on YouTube with this exact process. It explains how to use this formula to get the electric field vector in... It's unit vector notation, where you have the standard unit vectors i, j, and k. So if you want to see how to work out an example problem, type in electric field vector, r-hat, organic chemistry tutor in the YouTube search bar, and you're going to see how to use this formula.

So next time when you look at your physics textbook and you see this r-hat in the equations, you'll know what to do with it. By the way, for those of you who want the formula sheet with all these vectors, formulas, and more, feel free to check out the links in the description section below. There are more formulas that are associated with vectors. I didn't cover everything in this video. But if you want to see what those other formulas are, feel free to download that formula sheet for those of you who might be interested in it.

But I'm going to stop it here today for this video.

Transcript for:Understanding Vector Formulas in Physics

Transcript for:
Understanding Vector Formulas in Physics