https://youtu.be/e-L1memZH1w
PHYS 1101: Lecture Three, Part Seven
Okay. Let’s do an example. Let’s say we’re given these vectors, A and B, and we want to know what the magnitude is. First, we’re going to start out, do step one here, and then we’re going to find the scalar components. I’m going to copy vector A on the left. I’m going to copy vector B on the right, and I’m going to set up here to do my trig.
For vector A, I’m sketching in the two sides of this triangle. There’s Ax. There’s Ay. And here’s Ax. Let me do that in blue. Okay. So, that shows me the direction, and that will indicate the proper sign I need for these vectors, but I’m going to do my trigonometry just to get the values.
Let me sketch the triangle real quick again here, off to the side. The hypotenuse is 3. I have 45 degrees, and here is the sides Ay and Ax. So, Ax is going to be the hypotenuse, 3 times the cosine of 45 degrees. That’s 2.12. Ay is going to be 3 times the sine of 45 degrees. That’s also equal to 2.12.
Now, let me go off to the side of each of these and be sure I have the sign correct for the real scalar component. Ax, I actually need to make negative because the Ax vector component points to the left. So, that’s -2.12. Ay, the y component vector points up. That is in the positive y direction, based on my axes here that you see. So, Ay is going to stay the same, positive 2.12.
Okay. Let’s go take care of this for our B vector. I’m going to sketch my right triangle here, off to the right. Here’s going to be my y vector component, and here’s my x. This is the vector By. Here’s the vector Bx. So, now these arrow directions here will help me pick the right sign.
If I do my trigonometry, I know, let’s see, if this is 56 degrees, this has to be 90 degrees minus 56. That’s 34 degrees. So, I’ll go ahead. I have to get an angle, one of the angles inside of my right triangle, in order to do my trigonometry. So, I just had a tendency to want to go after this angle. So, given that 34 degrees, the hypotenuse is 6.5. I’m now set to get the lengths of the two sides.
Bx, that’s the opposite side to this angle, so it’s going to be 6.5 times the sine of 34 degrees. That is 3.6. Let me do my little arrow here. What’s the proper sign for this component? It has a magnitude of 3.6, but it does point to the negative x direction. So, I’m going to write that Bx is -3.6.
By, that involves the adjacent side to the angle that I have, so that’s my hypotenuse. 6.5 times the cosine of 34 degrees. The number I get for that is 5.4. Now, the real scalar component, after I carefully assign the proper sign to this By scalar component, I’m going to end up with -5.4. That vector points down.
I have my scalar components now. I’m ready to make my table. So, here I go. I’m going to do x and y. I’m going to do vector A plus vector B, is going to give me vector, we’ll call it “R”, the same resultant. Now, I just need to fill in my values from the table, the values from above, fill in my table.
So, let’s do for vector A. My x component is -2.12. My y component is +2.12. For Bx, I had -3.16. And y, I had -5.4. Now, I’m ready to just add down these columns. This means that my resultant for the x component is the sum of these two. That’s going to be -5.72. My y component is going to be the sum of those, which is going to be negative 3.28.
Let me emphasize again, this is the x scalar component of my resultant, and this is the y scalar component of my resultant. So, that means I can write off to the side here that my vector R is equal to -5.72 for its x scalar component, plus -3.28 for the ŷ unit vector. So, that’s my y scalar component. There’s the x, and there’s the y. So, that’s one way of writing this vector R.
Now, let’s go in and get the magnitude, as we were asked to do. I’m going to, real quick here, draw a sketch to be sure that my scalar values seem to make sense. But to do that, just to make it easier, I’m going to go up here and copy these vectors again. Let me paste them right here. Let me go in and just clean them up a bit here.
There’s vector A. Here’s vector A. Vector A plus vector B, the tail to tip sum would be from the initial tail out to the final tip. This is our vector R. And reminding you that our axes directions that we picked were the standard plus y up, plus x to the right. So, that means if I focus just on R, our resultant, and think of breaking it up into its components, I would expect to see a vector Ry that points down, and a vector Rx that points to the left. Ry, the scalar form of that should be a negative number, and Rx, the scalar component should also be a negative number.
And you know what? That’s consistent with what we found. Notice the relative size of these two is also consistent. The y scalar component is negative, but it’s a smaller magnitude than x, and that matches our picture.
So, what’s the magnitude of R? I just want to focus on the lengths of these two sides and use my Pythagorean to figure out the length of the hypotenuse here. So, I would write that my magnitude R is going to be the square root of just the lengths of the sides. So, here I don’t need to worry about the sine. I have Rx, which I’m just going to use 5.72, positive number squared plus my Ry squared, 3.28. Again, the sign just represents direction here. When I’m after the magnitude, I’m just doing geometry again. I’m comparing the lengths of the sides of this triangle, where for just this exercise, I’m not worried about directions. So, I can forget about the sign that conveys direction. When I multiply that out, I get 6.6.
Now, let’s say I wanted an angle. Just follow through here. I’ve redrawn it here, re-sketched it. Let’s say I want to go after this angle,