Lesson: Chapter - 4

Explanations

1.A

By adding A to B using the tip-to-tail method, we can see that (A) is the correct answer.

2.A

The vector 2A has a magnitude of 10 in the leftward direction. Subtracting B, a vector of magnitude 2 in the rightward direction, is the same as adding a vector of magnitude 2 in the leftward direction. The resultant vector, then, has a magnitude of 10 + 2 =12 in the leftward direction.

3.D

To subtract one vector from another, we can subtract each component individually. Subtracting the x-components of the two vectors, we get 3 –( –1) = 4, and subtracting the y-components of the two vectors, we get 6 – 5 = 1. The resultant vector therefore has an x-component of 4 and a y-component of 1, so that if its tail is at the origin of the xy-axis, its tip would be at (4,1).

4.D

The dot product of A and B is given by the formula A · B = AB cos ?. This increases as either A or B increases. However, cos ? ? = 0 when ? = 90°, so this is not a way to maximize the dot product. Rather, to maximize A · B one should set ? to 0º so cos ? = 1.

5.D

Let’s take a look at each answer choice in turn. Using the right-hand rule, we find that A × B is indeed a vector that points into the page. We know that the magnitude of A × B is AB sin ? , where ? is the angle between the two vectors. Since AB = 12, and since sin ?=1 , we know that A × B cannot possibly be greater than 12. As a cross product vector, is perpendicular to both A and B. This means that it has no component in the plane of the page. It also means that both A and B are at right angles with the cross product vector, so neither angle is greater than or less than the other. Last, B × A is a vector of the same magnitude as A × B , but it points in the opposite direction. By negating B × A , we get a vector that is identical to A × B .

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