Name: Cross Product
Uploaded: 2019-07-19
Duration: 4 min 16 s
Description: The cross product a × b is defined as a vector c that is perpendicular (orthogonal) to both a and b, with a particular direction and magnitude. Watch this vi...

Question

Find a unit vector perpendicular to each of the vectors

\vec{a} and \vec{b}

where

\vec{a} = 5 \hat{i} + 6 \hat{j} - 2 \hat{k} and \vec{b} = 7 \hat{i} + 6 \hat{j} + 2 \hat{k}

.

Accepted Answer

Given vectors: $\vec{a} = 5 \hat{i} + 6 \hat{j} - 2 \hat{k} and \vec{b} = 7 \hat{i} + 6 \hat{j} + 2 \hat{k}$

We know that, the unit vector perpendicular to the vectors $\vec{a} and \vec{b}$ is $\hat{n} = \pm (\frac{\vec{a} \times \vec{b}}{| \vec{a} \times \vec{b} |})$

Now, $\vec{a} \times \vec{b} = |\begin{array}{r} \hat{i} & \hat{j} & \hat{k} \\ 5 & 6 & - 2 \\ 7 & 6 & 2 \end{array}| = (12 + 12) \hat{i} - (10 + 14) \hat{j} + (30 - 42) \hat{k} = 24 \hat{i} - 24 \hat{j} - 12 \hat{k}$

And also, $|\vec{a} \times \vec{b}| = |24 \hat{i} - 24 \hat{j} - 12 \hat{k}| = \sqrt{24^{2} + {(- 24)}^{2} + {(- 12)}^{2}} = \sqrt{576 + 576 + 144} = \sqrt{1296} = 36$

Therefore, $\hat{n} = \pm \frac{(\vec{a} \times \vec{b})}{| \vec{a} \times \vec{b} |} = \pm \frac{24 \hat{i} - 24 \hat{j} - 12 \hat{k}}{36} = \pm (\frac{2}{3} \hat{i} - \frac{2}{3} \hat{j} - \frac{1}{3} \hat{k})$

Hence, the unit vector perpendicular to both $\vec{a} and \vec{b}$ is $\pm (\frac{2}{3} \hat{i} - \frac{2}{3} \hat{j} - \frac{1}{3} \hat{k})$ .

Find a unit vector perpendicular to each of the vectors $\vec{a} and \vec{b}$ where $\vec{a} = 5 \hat{i} + 6 \hat{j} - 2 \hat{k} and \vec{b} = 7 \hat{i} + 6 \hat{j} + 2 \hat{k}$ .

Important Questions on Vector Algebra

Learn and Prepare for any exam you want !

Find a unit vector perpendicular to each of the vectors a→ and b→ where a→=5i^+6j^-2k^ and b→=7i^+6j^+2k^.

Important Questions on Vector Algebra

Learn and Prepare for any exam you want !

Find a unit vector perpendicular to each of the vectors $\vec{a} and \vec{b}$ where $\vec{a} = 5 \hat{i} + 6 \hat{j} - 2 \hat{k} and \vec{b} = 7 \hat{i} + 6 \hat{j} + 2 \hat{k}$ .