Find a vector of magnitude 18 which is perpendicular to both the vectors 4i^-j^-3^ and -2i^+j^-2k^.

Answer 1

To find a vector that is perpendicular to both the given vectors, we can take their cross product.

Let A = 4i - j - 3k and B = -2i + j - 2k.

A x B = (Ai, Aj, Ak) x (Bi, Bj, Bk) = (Aj * Bk - Ak * Bj, Ak * Bi - Ai * Bk, Ai * Bj - Aj * Bi) = ((-1 * -2) - (-3 * 1), (-3 * -2) - (4 * -2), (4 * 1) - (-1 * -2)) = (2 - (-3), 6 - (-8), 4 - 2)

A x B = (5, 14, 2)

Now, we need to find a vector with magnitude 18 in the direction of the cross product.

First, let's find the magnitude of the cross product.

|A x B| = sqrt(5² + 14² + 2²) = sqrt(25 + 196 + 4) = sqrt(225)

|A x B| = 15

Now, let's find the unit vector in the direction of the cross product.

u = (5/15, 14/15, 2/15)

Finally, let's scale the unit vector to have a magnitude of 18.

v = 18 * u = (18 * 5/15, 18 * 14/15, 18 * 2/15) = (6, 16.8, 2.4)

So the vector with magnitude 18 that is perpendicular to both the given vectors is 6i + 16.8j + 2.4k.