If v = 2i and a = 7i - 9j, find a_N . A) 16 B) -16 D) 9 E) -9

Question

Quizplus · Accepted Answer

We have v = 2i and a = 7i - 9j. To find a_N, we first need to find the position vector of the point where v and a intersect. Let d be the displacement vector from the origin to this point. Then, since v and a are both tangent to the point, we have: d · v = 0 and d · a = 0 Expanding these dot products, we get: 2d1 = 0 (where d = d1i + d2j) 7d1 - 9d2 = 0 Solving for d1 and d2, we get: d1 = 0 d2 = 7/9 So the position vector of the point where v and a intersect is: d = (0)i + (7/9)j = (0, 7/9) Next, we need to find the projection of v onto the line passing through the origin and the point where v and a intersect. Let u be the unit vector in the direction of this line. Then, we have: u = (d - 0)/||d - 0|| = (0, 7/9)/||0, 7/9|| = (0, 1) (Note: ||v|| denotes the magnitude of vector v.) So u is simply the unit vector in the y-direction. To find the projection of v onto u, we use the dot product formula: proj_u v = (v · u) u Expanding, we get: v · u = (2i) · (0j + 1k) = 0 proj_u v = 0 So the projection of v onto the line passing through the origin and the point where v and a intersect is zero. Finally, we can compute a_N using the formula: a_N = (a · u) ||d - 0|| Expanding, we get: a · u = (7i - 9j) · (0j + 1k) = -9 ||d - 0|| = ||0, 7/9|| = 7/9 So: a_N = (-9) (7/9) = -7 Therefore, the answer is D) -9.

If V = 2i and a = 7i - 9j