Question

if a=2i+3j+k and b=3i+2j+4k,then find the value of (a+b)×(a-b)​

Answer 1

Explanation:

Given:

• $\sf{\overrightarrow{a}=2\hat{i}+3\hat{j}+\hat{k}}$
• $\sf{\overrightarrow{b}=3\hat{i}+2\hat{j}+4\hat{k}}$

To find:

• Value of $\sf{(\overrightarrow{a}+\overrightarrow{b})\times(\overrightarrow{a}-\overrightarrow{b})}$

Solution:

At first, find the value of $\sf{(\overrightarrow{a}+\overrightarrow{b})\:and\:(\overrightarrow{a}-\overrightarrow{b})}$.

$\star \: \sf \: ( \overrightarrow{a} + \overrightarrow{b}) \\ \\ = \sf \: (2 \hat{i} + 3 \hat{j} + \hat{k} + 3 \hat{i} + 2 \hat{j} + 4 \hat{k}) \\ \\ = \sf \: (5 \hat{i} + 5 \hat{j} + 5 \hat{k})$

&

$\star \: \sf \: ( \overrightarrow{a} - \overrightarrow{b}) \\ \\ = \sf \: \{(2 \hat{i} + 3 \hat{j} + \hat{k}) - (3 \hat{i} + 2 \hat{j} + 4 \hat{k}) \} \\ \\= \sf \: (2 \hat{i} + 3 \hat{j} + \hat{k}- 3 \hat{i} - 2 \hat{j} - 4 \hat{k}) \\ \\ \sf = ( - \hat{i} + \hat{j} - 3 \hat{k})$

Now find the value of $\sf{(\overrightarrow{a}+\overrightarrow{b})\times(\overrightarrow{a}-\overrightarrow{b})}$.

$\sf{(\overrightarrow{a}+\overrightarrow{b})\times(\overrightarrow{a}-\overrightarrow{b})} \\ \\ = \sf \: (5 \hat{i} + 5 \hat{j} + 5 \hat{k}) \times ( - \hat{i} + \hat{j} - 3 \hat{k})\\ \\=\sf\left|\begin{array}{ccc} \ \ \hat{i} & \ \ \hat{j} & \ \ \hat{k} \\ \ \ 5 & \ \ 5 & \ \ 5 \\ -1 & \ \ 1 & - 3\end{array}\right|\\ \\ = \sf \: \hat{i}( - 15 - 5) + \hat{j}( - 5 + 15) + \hat{k}(5 + 5) \\ \\ \sf \: = - 20 \hat{i} + 10 \hat{j} + 10 \hat{k}$

Hence, the value of $\sf{(\overrightarrow{a}+\overrightarrow{b})\times(\overrightarrow{a}-\overrightarrow{b})}$ is $\sf{- 20 \hat{i} + 10 \hat{j} + 10 \hat{k}}$