Question

For what values of a are v = a i − j + k and w = a i + a j − 3 k perpendicular? give exact values.

Answer 1

If the two vectors are perpendicular, then their dot product is zero.

`\vec v \cdot \vec w = (a\,\vec\imath - \vec\jmath + \vec k) \cdot (a\,\vec\imath + a\,\vec\jmath - 3\,\vec k) = a^2 - a - 3 = 0`

Solve for
`a`. With the quadratic formula, we find

`a = \frac{1 \pm √(13)}2`