In this video, we're going to talk about vectors. So looking at this question, which of the following quantities is not a vector? Would you say displacement, velocity, acceleration, mass, or force? We need to be familiar with two things.
You need to be familiar with scalar quantities. and vector quantities. A scalar quantity is something that has magnitude but no direction. For instance, temperature is a scalar quantity. It has a magnitude, let's say if it's 80 degrees Fahrenheit, that's the magnitude of the temperature, but you can't apply direction to temperature.
You can't say it's, I mean you could say it, but it doesn't make any sense. If you were to say it's 80 degrees Fahrenheit east, That wouldn't be relevant. Force, for instance, is a vector. Because you can describe it using magnitude and direction. For instance, you can have a force of 100 newtons directed at an angle of 30 degrees above the x-axis.
So this is the magnitude of the force. and this is the direction of the force which makes it a vector. So anything that has both magnitude and direction is a vector. So now that we know that force is a vector, we can eliminate answer choice E.
What else do we need to know? You need to know that distance is scalar but displacement is a vector. If you were to say a person ran 45 meters, you're describing a person's distance because you didn't apply direction to it. But if you were to say a person ran 45 meters east, you're now describing the displacement of the person and not its distance.
So displacement is basically distance with direction. So we can eliminate answer choice A. Now you also need to know that speed is a scalar quantity, and velocity is a vector quantity.
So just as displacement is distance with direction, velocity is speed with direction. Velocity tells you how fast you're going and where you're going. Speed simply tells you how fast you're going. And by the way, displacement is the change in position. acceleration tells you how fast the velocity is changing acceleration is also a vector so for this problem the correct answer is answer choice D mass is a scale of quantity For instance, let's say an object has a mass of 10 kilograms.
You won't say that the object has a mass of 10 kilograms north. It wouldn't make sense. So mass is a scale of quantity. It has magnitude only, but no direction.
Now consider this problem. A force vector has a magnitude of 100 newtons directed at an angle of 30 degrees above the x-axis. Calculate the magnitude of the x and y components of this force vector.
So first, let's write some equations. So let's say we have the force vector F. We can break it up into its x component and its y component.
Let's call this angle theta. Now let's review some things that you might have learned in trigonometry if you've taken that class. There's something called SOHCAHTOA. Let's focus on the SOH part of SOHCAHTOA. The S stands for sine.
Sine of the angle is equal to, the O represents the opposite side, H is the hypotenuse. So opposite to the angle theta is Fy. So let's write opposite. Adjacent to the angle theta is Fx.
And then across the 90 degree angle, that is the hypotenuse, which is F in this case. So using the formula, sine theta is opposite over hypotenuse. We can say that sine theta is going to be Fy over F. Now, if you were to rearrange this equation, if you were to multiply both sides by F, you'll get that the y component of the force vector is the magnitude of f times sine of the angle theta with respect to the x-axis. So that's the first equation that you need to be aware of.
Now let's consider the second equation in the Ka part of SOHCAHTOA. So c stands for cosine. Cosine of the angle is going to equal the adjacent side, which is Fx over the hypotenuse which we know it to be F Now if we do the same thing if we were to multiply both sides by F We'll get that the x component of the force vector is the magnitude of F times cosine of the angle Now for the last one Toa tangent theta is equal to O or opposite which is FY over the adjacent part which is FX. So I like to use this formula to calculate the angle because in some problems you need to determine the magnitude and the angle. To determine the angle you need to take the arctangent or the inverse tangent.
If we were to take the arc tan of both sides of the equation, the arc tan and the tan will cancel on the left. So we'll get that the angle is equal to arc tan fy over fx. So these are some things you want to write down, because it's going to be very helpful.
Particularly when you're solving problems later in this video. Now let's go back to a right triangle. Let's say this is A, B, and C.
According to the Pythagorean theorem, we know that C squared is equal to A squared plus B squared. Well, C is the hypotenuse, so we can replace C with F. A corresponds to F of X in this example. and b corresponds to f of y. So if we wish to calculate the magnitude of a vector, and we know the x and y components, it's simply going to be the square root of f sub x squared plus f sub y squared.
So make sure you're familiar with these four formulas. Now, let's go ahead and finish this problem. So first, let's draw a picture.
So we have a force vector that is directed at an angle of 30 degrees above the x-axis. So here we have the x-axis, and this is the y-axis. So it would be somewhere in that area.
And the magnitude of this force vector is 100 Newtons. So let's break it up into its x and y components. So this is going to be the x component of the vector, and this is the y component of that vector. And then we have our angle here, which is 30 degrees.
So with this information, go ahead and calculate the magnitude of the x and the y components of this force vector. So we know that f of x is equal to f cosine theta. f is 100, and theta is 30. Now, cosine of 30 degrees, that's equal to the square root of 3 over 2. So we have 100 divided by 2, which is 50. So the exact answer is 50 square root 3. Now for those of you who want a decimal value, if you multiply 50 and the square root of 3, you're going to get 86.6 newtons. So that's the value of F sub X. We'll write it as 86.6 newtons.
Now let's do the same for the Y component. This is going to be f times sine theta. So f is 100, and then times sine of 30. Sine 30 is 1 half.
So half of 100 is 50. So we can say that f sub y is equal to 50 degrees, I mean, not 50 degrees. So 50 newtons. That's the unit of force. Now we're told to express the answer using standard unit vectors. But you might be wondering, what is a unit vector?
A unit vector is simply a vector with a length or magnitude of 1. Now we want to express it using the unit vectors i, j, and k. So let's draw a three-dimensional coordinate system, where this is z, this is x, and this is y. I is a unit vector along the x-axis, so it has a length of 1. J is a unit vector along the y-axis, and k is a unit vector along the z-axis. So what we need to know is that...
The unit vector I is associated with the X component, the unit vector J is associated with the Y component, and the unit vector K is associated with the Z component. So to express the answer using standard unit vectors, we can say that the original force vector F is equal to 86.6 times the unit vector i because that's the x component and then plus 50 times j which tells us that the magnitude of the y component is 50. so this is one way in which we can express the force vector so you can express a force vector or any vector using the magnitude the angle or you can express it in component form using the X and Y components