Final answer:
To find a unit vector orthogonal to vectors a and b, calculate the cross product of the two vectors and normalize the result to unit length. The resulting unit vector orthogonal to both a and b is (5/√123)i - (7/√123)j - (7/√123)k.
Step-by-step explanation:
To find a unit vector that is orthogonal to both vector a and vector b, we can use the cross product. The cross product of two vectors results in a vector that is orthogonal to both of the original vectors.
The vectors given are:
- Vector a = ⟨−3, −2, −3⟩
- Vector b = ⟨−2, 1, −1⟩
To calculate the cross product, we can use the determinant of a matrix:
A × B =
| i j k |
|Ax Ay Az|
|Bx By Bz|
For vectors a and b, we get:
A × B =
| i j k |
| -3 -2 -3 |
| -2 1 -1 |
= i(2 × -1 - (-3) × 1) - j(-3 × -1 - (-2) × -2) + k(-3 × 1 - (-2) × -2)
= i(2 + 3) - j(3 + 4) + k(-3 - 4)
= 5i - 7j - 7k
The resulting vector is not yet a unit vector. To make it a unit vector, we need to divide it by its magnitude.
The magnitude of the cross product is:
|A × B| = √(5² + (-7)² + (-7)²) = √(25 + 49 + 49) = √123
The unit vector is:
û = (1/√123) × (5i - 7j - 7k)
= (5/√123)i - (7/√123)j - (7/√123)k