Name: Parallelogram Law for Addition of Two Vectors
Uploaded: 2019-07-19
Duration: 5 min 1 s
Description: Now, we know one of the methods to add the two vectors. It is time to learn another law to add the two vectors. Explore this video to learn about it.

Question

Find the unit vector perpendicular to the vectors $2 \hat{i} - 3 \hat{j} + \hat{k}$ , and $- \hat{i} + 2 \hat{j} - \hat{k}$

Accepted Answer

We know that Cross Product of two vectors is a vector perpendicular to both the vectors.

Thus, vectors perpendicular to both the vectors $2 \hat{i} - 3 \hat{j} + \hat{k}$ , and $- \hat{i} + 2 \hat{j} - \hat{k}$ are $\pm [(2 \hat{i} - 3 \hat{j} + \hat{k}) \times (- \hat{i} + 2 \hat{j} - \hat{k})]$

We know that if $\vec{a} = a_{1} \hat{i} + a_{2} \hat{j} + a_{3} \hat{k}$ and $\vec{b} = b_{1} \hat{i} + b_{2} \hat{j} + b_{3} \hat{k}$ then, $\vec{a} \times \vec{b} = |\begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ a_{1} & a_{2} & a_{3} \\ b_{1} & b_{2} & b_{3} \end{matrix}|$

Therefore,

$(2 \hat{i} - 3 \hat{j} + \hat{k}) \times (- \hat{i} + 2 \hat{j} - \hat{k}) = |\begin{array}{l} \hat{i} & \hat{j} & \hat{k} \\ 2 & - 3 & 1 \\ - 1 & 2 & - 1 \end{array}|$

$= \hat{i} (3 - 2) - \hat{j} (- 2 + 1) + \hat{k} (4 - 3)$

$= \hat{i} + \hat{j} + \hat{k}$

Thus, $\pm (\hat{i} + \hat{j} + \hat{k})$ are the vectors perpendicular to the vectors $2 \hat{i} - 3 \hat{j} + \hat{k}$ and $- \hat{i} + 2 \hat{j} - \hat{k}$

So, the unit vectors perpendicular to both the given vectors are $\pm (\frac{\hat{i} + \hat{j} + \hat{k}}{|\hat{i} + \hat{j} + \hat{k}|})$

Now, $|\hat{i} + \hat{j} + \hat{k}| = \sqrt{{(1)}^{2} + {(1)}^{2} + {(1)}^{2}} = \sqrt{3}$

Hence, $\pm \frac{1}{\sqrt{3}} (\hat{i} + \hat{j} + \hat{k})$ are unit vectors perpendicular to both the given vectors.

Find the unit vector perpendicular to the vectors $2 \hat{i} - 3 \hat{j} + \hat{k}$ , and $- \hat{i} + 2 \hat{j} - \hat{k}$

Important Questions on Vectors

Learn and Prepare for any exam you want !

Find the unit vector perpendicular to the vectors 2i^-3j^+k^, and -i^+2j^-k^

Important Questions on Vectors

Learn and Prepare for any exam you want !

Find the unit vector perpendicular to the vectors $2 \hat{i} - 3 \hat{j} + \hat{k}$ , and $- \hat{i} + 2 \hat{j} - \hat{k}$