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 subx and it has a y component V suby 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 sin 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 mag magnitude of vector v for a two-dimensional Vector is vx^ 2 plus v y^2 so you get this formula by using the Pythagorean theorem A2 + B2 = c^2 when you solve for C it's the square < TK of a s + b s 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^ 2 plus v y^ 2 plus vz^ 2 so that's if you have a 3D Vector now the angle the angle Theta in this triangle is need to find the direction of the vector for a 2d vector it's AR tangent VY over VX now vector v can be written like this it could be written in terms of its components VX I plus VY J if you have a 2d Vector for 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 time cosine Theta I plus the magnitude of V time sin 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 one 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 standard unit Vector I is 1 comma 0 comma 0 it basically has a length of one in the X Direction This is x y z the standard unit Vector J is 0 1 0 so it has a length of one in the y direction and the standard unit vector k is 0 0 1 it has a length of one in a 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 xaxis this is the Y AIS and this is the z-axis and let's say this is the L of one this is a length of one and this is a length of one this would be the unit Vector I this will be J and this is K so they all have a magnitude of one 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 Scala quantity something that's not a vector the formula for the dot product is you take the magnitude of a multiply by the magnitude of B and multiply 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 a z * bz now just like the magnitude of V the magnitude of a is equal to the square root of ax^2 + a y^ 2 + a z^ 2 for 3D Vector for 2D Vector just take out the Z and it's just going to be the otk of ax2 + a y^ 2 now you would find b in a the similar fashion it's the square root of bx^ 2 + b y^ 2 plus bz ^ 2 now a good example of the do product formula is the calculation of work work is force time displacement time 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° angles the work done by this force is zero because cosine 90 is zero when the force and the displacement Vector are parallel to the to each other then the work done is going to be the maximum that that force can deliver because cosine 0 is one so it's just going to be f * D so when you multiply two vectors using the do product formula you're going to get a scal of quantity so that's the scalar do 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 sin 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 a y a z and then BX b y 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 scal of quantity for the cross product formula 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 a print out of this formula sheet or the formulas I'm going over feel free to check out the links in the description section below now if you 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 if 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 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 Time the X component of b time the X component of a Time the Z component of b so the fact that you see this 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 multipli by the Y component of b minus the 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 do 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 scal 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 our 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 * B * sin Theta without the unit Vector but because I don't have the absolute value 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 it 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 Time s 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 torqus another example is the magnetic force on a moving charge so let's say if you have a 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 his 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 in the X Direction it could be in a y direction it can 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 a scale of quanty so we won't have an error on top of 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 time the magnetic field time q and then just sin Theta but now if you want the entire Vector itself it's V cross B * Q * sin 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 time R hat a lot of formulas in physics with Calculus if you're taking that course you'll see they'll Define a vector usually in this format it's magnitude time 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 unit vect I mean the position Vector R to define the position Vector R we need two points the first point ideally would be the origin so it will be easy 0 0 0 the second point B uh will have the points XB YB ZB a will be x a y a z a but to determine the value of let me color code this to determine the value of the position Vector is 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 y a which if a is the origin ya a is just zero and this is times unit Vector J and then this is going to be ZB minus Za times unit vector k so this is rx I and then this is r y j and this is RZ k 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 rx2 + r y^ 2 + r z^ 2 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 the moving charge is the cross product Vector formula V cross B * Q * sin th 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 you know the value of that Vector with both magnitude and Direction included another example of this formula is the electric field Vector any 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 time Q the magnitude of this charge / R 2 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 e x e y if it's a 2d vector and then e z as well as 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 its 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 uh for those of you who might be interested in it but I'm going to stop it here today for this video