Question

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Answer 1

Given:

The two vectors are:

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

$\overrightarrow{b}=\hat{i}-3\hat{j}+5\hat{k}$

To find:

The value of $|\overrightarrow{a}\times \overrightarrow{b}|$.

Solution:

We have,

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

$\overrightarrow{b}=\hat{i}-3\hat{j}+5\hat{k}$

The cross product of these two vectors is:

$\overrightarrow{a}\times \overrightarrow{b}=\begin{vmatrix}\hat{i}&\hat{j}&\hat{k}\\2&-1&1\\1&-3&5\end{vmatrix}$

$\overrightarrow{a}\times \overrightarrow{b}=\hat{i}[(-1)(5)-(1)(-3)]-\hat{j}[(2)(5)-(1)(1)]+\hat{k}[(2)(-3)-(-1)(1)]$

$\overrightarrow{a}\times \overrightarrow{b}=\hat{i}[-5+3]-\hat{j}[10-1]+\hat{k}[-6+1]$

$\overrightarrow{a}\times \overrightarrow{b}=-2\hat{i}-9\hat{j}-5\hat{k}$

Now the magnitude of the cross product is:

$|\overrightarrow{a}\times \overrightarrow{b}|=\sqrt{(-2)^2+(-9)^2+(-5)^2}$

$|\overrightarrow{a}\times \overrightarrow{b}|=\sqrt{4+81+25}$

$|\overrightarrow{a}\times \overrightarrow{b}|=\sqrt{110}$

Therefore, the value of $|\overrightarrow{a}\times \overrightarrow{b}|$ is $\sqrt{110}$.