# How do you find a unit vector u that is orthogonal to a and b where a = 7 i - 5 j + k and b = -7 j - 5 k?

Nov 9, 2016

$\hat{c} = \frac{1}{5 \sqrt{158}} \left(- 18 \hat{i} + 35 \hat{j} - 49 \hat{k}\right)$

#### Explanation:

The cross product of the two given vectors will give you a vector that is orthogonal; to make it a unit vector, you merely divide it by its magnitude.

barc = bara xx barb = | (hati, hatj, hatk, hati, hatj), (7,5,1,7,5), (0,-7,-5,0,-7) | =

$\overline{c} = \hat{i} \left\{\left(5\right) \left(- 5\right) - \left(1\right) \left(- 7\right)\right\} + \hat{j} \left\{\left(1\right) \left(0\right) - \left(7\right) \left(- 5\right)\right\} + \hat{k} \left\{\left(7\right) \left(- 7\right) - \left(5\right) \left(0\right)\right\} =$

$\overline{c} = - 18 \hat{i} + 35 \hat{j} - 49 \hat{k}$

The above vector, $\overline{c}$ is orthogonal to both $\overline{a} \mathmr{and} \overline{b}$.

To make $\overline{c}$ a unit vector, compute its magnitude and then divide the vector by it:

$| \overline{c} | = \sqrt{{\left(- 18\right)}^{2} + {35}^{2} + {\left(- 49\right)}^{2}}$

$| \overline{c} | = \sqrt{324 + 1225 + 2401}$

$| \overline{c} | = \sqrt{3950} = 5 \sqrt{158}$

The unit vector is:

$\hat{c} = \frac{1}{5 \sqrt{158}} \left(- 18 \hat{i} + 35 \hat{j} - 49 \hat{k}\right)$