4. Add and subtract vectors.
Hold onto your hats, because this is a lot of stuff. The good news is that it's all lumped into one standard because it all involves the same basic concepts, so this isn't going to be nearly as intimidating as it might appear from the title.
If v = <3, 2> and w = <4, 7> find v + w.
Remember, the first number in the ordered pair is the horizontal distance, the second the vertical. To find the resultant, add the horizontal components first, then the vertical. So we end up with 3 + 4 = 7 and 2 + 7 = 9. The vector that matches it is (B).
If v = <5, 2> and w = <8, 2>, find v – w.
The additive inverse of w just means changing the signs of both components. If we do that, we'll get <-8, -2> for -w. When we add that to v = <5, 2>, we get <-3, 0>.
Why is it easier to add vectors of the same direction?
If two vectors being added are going the same direction, all we have to do is add the magnitudes to get the length of the new vector. No complicated Law of Cosines necessary. Just add the two together, and we're golden.
If vector v = <1, 3> and w = <9, -2>, what is the direction when the vectors are added?
Adding the vectors means adding their components. Once we do that, we have <10, 1> for v + u. The direction of a vector is essentially its slope, and we can figure that out using the linear mantra "rise over run." Since the second component is the vertical one, that goes on top, while the first component goes on the denominator. That means our answer is (B).
The vector resulting from v – w is the same as the vector resulting from which of the following vectors?
Subtracting one vector from another is like adding the opposite vector. So v – w is the same as v + (-w). That's (C), but what about (A) and (B)? Those two answer choices would have us believe that v – w is the same as w – v, which is untrue even for numbers. (The number 2 – 1 = 1 isn't the same as 1 – 2 = -1.)
Vectors v and w are pointing in the same direction. Which of the following is true?
It's not possible to add vectors (v and w), which have direction, and get only numbers (||v|| or ||v + w||) as a result, so (A) and (C) are automatically out. If v and w are pointing in the same direction, the sum of their magnitudes equals the magnitude of their sum (reread that and double-check it if you need to).
What about (B)? Well, the magnitude of -w is the same as the magnitude of w. That means ||v|| – ||-w|| = ||v|| – ||w||. We know that ||v|| – ||w|| isn't the same as ||v|| + ||w||, so (B) can't be right.
Two vectors have the same magnitude, a. Which of the following is true about the magnitude of the resulting vector if the two vectors are added together?
At first glance, it seems obvious that the two vectors will add up to twice the length of the original vector. However, the directions of the vectors aren't mentioned at all. That means the vectors could be facing the same direction or not—we just don't know. That would change which approach we use to calculate the magnitude, so there's no way we can be sure of the magnitude.
Two vectors have the same magnitude, a, and opposite directions. Which of the following is true about the magnitude of the resulting vector if the two vectors are subtracted?
If one of the vectors is a, then the other vector is -a (since the two vectors face opposite directions). If the two vectors are subtracted, we have a – -a, which is really just 2a. The magnitude of 2a is exactly what you'd think it'd be. Yup, it's 2a. If we added the two vectors instead of subtracted, our answer would be (C), but since we subtracted them, we have (A) as our answer.
Two vectors of magnitudes 6 and 8 respectively, have an angle of 30° between them. Find the magnitude of the resultant vector.
If we draw a parallelogram with the two vectors starting out at the same initial point, we can label the sides 6 and 8. If we use the Law of Cosines, c2 = a2 + b2 – 2ab cos(C), plugging in 6 for a and 8 for b, we have . That'll give us approximately 13.5 for c.
Two vectors of 9 and 12 respectively, form an angle of 80 degrees. Find the magnitude of the resultant vector.
Since the vectors aren't facing the same direction, we have to make use of the parallelogram rule. If we use the Law of Cosines again, we'll end up with c2 = 81 + 144 – 216(-0.1736), which means c is approximately 16.2. That means the magnitude of the resultant vector is 16.2.