Question

Find a unit vector that is orthogonal to both u and v. Given u = (3, -5, 4) and v = (1, 2, 5).

Answer 1

To find a unit vector orthogonal to vectors u and v, calculate the cross product of u and v, then divide this vector by its magnitude to normalize it into a unit vector.

Step-by-step explanation:

To find a unit vector that is orthogonal to both vectors u and v, where u = (3, -5, 4) and v = (1, 2, 5), we can use the cross product. The cross product of two vectors gives a new vector that is orthogonal to both original vectors. The steps to find the unit vector orthogonal to u and v are as follows:

1. Compute the cross product w = u × v.
2. Calculate the magnitude of vector w.
3. Divide each component of w by its magnitude to get the unit vector.

Now let's perform these steps:

1. Calculate the cross product:
   w = u × v = (3 × 5 - (-5) × 2, -3 × 1 + 4 × 2, 3 × 2 - (-5) × 1) = (15 + 10, -3 + 8, 6 + 5) = (25, 5, 11).
2. Find the magnitude of w:
   |w| = √(25² + 5² + 11²) = √(625 + 25 + 121) = √(771).
3. Divide each component by the magnitude to obtain the unit vector:
   unit w = × (25/√771, 5/√771, 11/√771).

This gives us the unit vector orthogonal to both u and v.