Question

If a=i+2j-3k b=3i-j+2k then show that a+b a-b are mutually perpendicular

Answer 1

$\textbf{Concept:}$

$\text{Two vectors }\overrightarrow{a}\text{ and }\overrightarrow{b}\text{ are perpendicular if and only if }\overrightarrow{a}.\overrightarrow{b}=0$

$\textbf{Given:}$

$\overrightarrow{a}=\overrightarrow{i}+2\overrightarrow{j}-3\overrightarrow{k}$

$\overrightarrow{b}=3\overrightarrow{i}+\overrightarrow{j}+2\overrightarrow{k}$

$\implies\,\overrightarrow{a}+\overrightarrow{b}=4\overrightarrow{i}+3\overrightarrow{j}-\overrightarrow{k}$ and

$\implies\,\overrightarrow{a}-\overrightarrow{b}=-2\overrightarrow{i}+\overrightarrow{j}-5\overrightarrow{k}$

$(\overrightarrow{a}+\overrightarrow{b}).(\overrightarrow{a}-\overrightarrow{b})=(4\overrightarrow{i}+3\overrightarrow{j}-\overrightarrow{k}).(-2\overrightarrow{i}+\overrightarrow{j}-5\overrightarrow{k})$

$(\overrightarrow{a}+\overrightarrow{b}).(\overrightarrow{a}-\overrightarrow{b})=4(-2)+3(1)+(-1)(-5)$

$\implies(\overrightarrow{a}+\overrightarrow{b}).(\overrightarrow{a}-\overrightarrow{b})=-8+3+5=0$

$\therefore\overrightarrow{a}\perp\overrightarrow{b}$

That is,they are mutually perpendicular to each other

Answer 2

Answer:

Step-by-step explanation:

Given data, $\vec{a} = i + 2j -3k , \vec{b} = 3i-j+2k$

Two vectors to be perpendicular if,

$\vec{a}\times\vec{b} = 0$

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

⇒ $[tex]\vec{a} + \vec{b} = (i + 2j -3k) + (3i-j+2k)$

                                   $= (4i + j -k)$

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

                              $=(- 2i + 3j - 5k)$

$(\vec{a} + \vec{b})\times(\vec{a} - \vec{b}) = (4i + j -k)\times(- 2i + 3j - 5k)$

= - 8 + 3 + 5

= 0

It means$\vec{a}$and $\vec{b}$is perpendicular.