Name: Types of Vectors
Uploaded: 2019-07-19
Duration: 3 min 35 s
Description: A vector is a quantity that has both magnitude and direction. Did you know that there are several types of vectors? Watch the video to learn about them.

Question

Find a unit vector perpendicular to each of the vectors

(\vec{a} + \vec{b})

and

(\vec{a} - \vec{b})

, where

\vec{a} = \hat{i} + \hat{j} + \hat{k}, \vec{b} = \hat{i} + 2 \hat{j} + 3 \hat{k}

.

Accepted Answer

The given two vectors are $\vec{a} = \hat{i} + \hat{j} + \hat{k}$ and $\vec{b} = \hat{i} + 2 \hat{j} + 3 \hat{k}$ .

So, $(\vec{a} + \vec{b}) = (\hat{i} + \hat{j} + \hat{k}) + (\hat{i} + 2 \hat{j} + 3 \hat{k})$ $= 2 \hat{i} + 3 \hat{j} + 4 \hat{k}$

And $(\vec{a} - \vec{b}) = (\hat{i} + \hat{j} + \hat{k}) - (\hat{i} + 2 \hat{j} + 3 \hat{k})$ $= - \hat{j} - 2 \hat{k}$

A vector which is perpendicular to both $(\vec{a} + \vec{b})$ and $(\vec{a} - \vec{b})$ is $(\vec{a} + \vec{b}) \times (\vec{a} - \vec{b})$

So, $(\vec{a} + \vec{b}) \times (\vec{a} - \vec{b}) = |\begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 2 & 3 & 4 \\ 0 & - 1 & - 2 \end{matrix}|$

$= - 2 \hat{i} + 4 \hat{j} - 2 \hat{k}$

And $|(\vec{a} + \vec{b}) \times (\vec{a} - \vec{b})| = \sqrt{(- 2)^{2} + (4)^{2} + (- 2)^{2}}$ $= \sqrt{24} = 2 \sqrt{6}$

So, the unit vector perpendicular to both $(\vec{a} + \vec{b})$ and $(\vec{a} - \vec{b})$ is

$= \pm \frac{(\vec{a} + \vec{b}) \times (\vec{a} - \vec{b})}{|(\vec{a} + \vec{b}) \times (\vec{a} - \vec{b})|}$

$= \pm (\frac{- 2 \hat{i} + 4 \hat{j} - 2 \hat{k}}{2 \sqrt{6}})$

$= \pm (\frac{- \hat{i} + 2 \hat{j} - \hat{k}}{\sqrt{6}})$

Which is the required unit vector.

Find a unit vector perpendicular to each of the vectors $(\vec{a} + \vec{b})$ and $(\vec{a} - \vec{b})$ , where $\vec{a} = \hat{i} + \hat{j} + \hat{k}, \vec{b} = \hat{i} + 2 \hat{j} + 3 \hat{k}$ .

Important Questions on Vector Algebra

Learn and Prepare for any exam you want !

Find a unit vector perpendicular to each of the vectors (a→+b→) and (a→-b→), where a→=i^+j^+k^,b→=i^+2j^+3k^.

Important Questions on Vector Algebra

Learn and Prepare for any exam you want !

Find a unit vector perpendicular to each of the vectors $(\vec{a} + \vec{b})$ and $(\vec{a} - \vec{b})$ , where $\vec{a} = \hat{i} + \hat{j} + \hat{k}, \vec{b} = \hat{i} + 2 \hat{j} + 3 \hat{k}$ .