Question

Find a unit vector perpendicular to the vectors given A=4i+2j+2k and B=4i-4j+8k.

Answer 1

To find a unit vector perpendicular to the given vectors A and B, we can take the cross product of the two vectors.

Step-by-step explanation:

To find a unit vector perpendicular to the vectors A=4i+2j+2k and B=4i-4j+8k, we can take the cross product of the two vectors. The cross product of two vectors A and B, denoted as A x B, is a vector that is perpendicular to both A and B. The magnitude of the cross product is given by |A x B| = |A||B|sin(theta), where theta is the angle between A and B.

Using this formula, we can calculate the cross product A x B and then divide it by its magnitude to obtain the unit vector. Therefore, the unit vector perpendicular to the given vectors A and B is (2/3)i - (1/3)j - (2/3)k.

Answer 2

The cross product of A and B is perpendicular to both A and B.

A × B = (4i + 2j + 2k) × (4i - 4j + 8k)

A × B = 16 (i × i) - 16 (i × j) + 32 (i × k) + 8 (j × i) - 8 (j × j) + 16 (j × k) + 8 (k × i) - 8 (k × j) + 16 (k × k)

A × B = -16 (i × j) - 32 (k × i) - 8 (i × j) + 16 (j × k) + 8 (k × i) + 8 (j × k)

A × B = -16k - 32j - 8k + 16i + 8j + 8i

A × B = 24i - 24j - 24k

The magnitude of A × B is

||A × B|| = 24 ||i - j - k|| = 24√3

Dividing A × B by its magnitude gives a unit vector,

(A × B)/||A × B|| = 1/√3 (i - j - k)