Question

Find the length and direction (when defined) of u times v and v times u.

u = 9i- 2j - 8k,
v = 8i-8k

Answer 1

To find the length and direction of the cross product of u and v, we can use the cross product formula. The length of u times v is 16sqrt(21) and the direction is (-i + j - 4k) / sqrt(21). The length of v times u is 96 and the direction is (2i + 2j + k) / sqrt(3).

Step-by-step explanation:

To find the length and direction of u times v and v times u, we can use the cross product formula. The cross product of two vectors u and v is given by |u x v| = |u|*|v|*sin(theta), where theta is the angle between u and v. For u times v, u = 9i - 2j - 8k and v = 8i - 8k. The cross product is calculated as follows:

1. Calculate the cross product: u x v = (9 * 0 - (-2) * 8) i - (9 * 8 - (-2) * 8) j + (9 * (-8) - (-2) * 8) k = -16i + 16j - 64k
2. The length (magnitude) of u x v is |u x v| = sqrt((-16)^2 + 16^2 + (-64)^2) = sqrt(256 + 256 + 4096) = sqrt(5376) = 16sqrt(21)
3. The direction (when defined) of u x v is the unit vector in the direction of u x v, which is (-16i + 16j - 64k) / |u x v| = (-i + j - 4k) / sqrt(21)

For v times u, the cross product is calculated in the same way: v x u = (8 * 0 - (-8) * (-8)) i - (8 * (-8) - (-8) * 0) j + (8 * (-2) - (-8) * (-2)) k = 64i + 64j + 32k. The length of v x u is |v x u| = sqrt(64^2 + 64^2 + 32^2) = sqrt(4096 + 4096 + 1024) = sqrt(9216) = 96. The direction, when defined, is the unit vector in the direction of v x u, which is (64i + 64j + 32k) / |v x u| = (2i + 2j + k) / sqrt(3).