Question

If ⃗ = + + is perpendicular to both ⃗ = 5 + − 2 and = 3 − 3 + 6 , find and .

Answer 1

The values of
`m` and
`n` are 0 and 2, respectively.

Explanation:

If
`\vec {c} = m\,\hat{i} + n\,\hat{j} + \hat{k}` is perpendicular to
`\vec {a} = 5\,\hat{i} + \hat{j} -2\,\hat{k}` and
`\vec {b} = 3\,\hat{i} - 3\,\hat{j} + 6\,\hat{k}`, then the following relationships must be observed:

`\vec {c}\,\bullet\,\vec {a} = 0` (1)

`\vec{c}\,\bullet \,\vec{a} = 0` (2)

Then, we expand the previous expressions:

`(m, n, 1)\,\bullet\,(5, 1, -2) = 0`

`5\cdot m + n = 2` (1b)

`(m, n, 1)\,\bullet\,(3, -3, 6) = 0`

`3\cdot m - 3\cdot n = -6` (2b)

Then, we solve for
`m` and
`n`:

`m = 0, n = 2`

The values of
`m` and
`n` are 0 and 2, respectively.