If a =2 i - j -2 k and b =7 i +2 j -3 k , then express b in the f… - Vidyadip

Question

If $ \vec{\text{a}}=2\hat{\text{i}}-\hat{\text{j}}-2\hat{\text{k}}\ \text{and}\ \vec{\text{b}}=7\hat{\text{i}}+2\hat{\text{j}}-3\hat{\text{k}}, $ then express $ \overrightarrow{\text{b}}$ in the form of $\overrightarrow{\text{b}}=\ \overrightarrow{\text{b}}_1+\overrightarrow{\text{b}}_2,$ where $ \overrightarrow{\text{b}}_1$ is parallel to $\overrightarrow{\text{a}}$ and $ \overrightarrow{\text{b}}_2$ is perpendicular to$\overrightarrow{\text{a}}$.

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Answer

$\overrightarrow{\text{b}}_1$||$\overrightarrow{\text{a}}$ $\Rightarrow\text{let}$ $\overrightarrow{\text{b}}_1$= $\lambda$ $(2\hat{\text{i}}-\hat{\text{j}}-2\hat{\text{k}})$
$\overrightarrow{\text{b}}_2$= $\overrightarrow{\text{b}}$- $\overrightarrow{\text{b}}_1$= $(7\hat{\text{i}}+2\hat{\text{j}}-3\hat{\text{k}})-(2\lambda\hat{\text{i}}-\lambda\hat{\text{j}}-2\lambda\hat{\text{k}})$
= $(7-2\lambda)\hat{\text{i}}+(2+\lambda)\hat{\text{j}}-(3-2\lambda)\hat{\text{k}}$
$\overrightarrow{\text{b}}_2\perp\overrightarrow{\text{a}}\Rightarrow2(7-2\lambda)-1(2+\lambda)+2(3-2\lambda)$
$\Rightarrow\lambda=2$
$\therefore\ \vec{\text{b}_1}=\ 4\hat{\text{i}}-2\hat{\text{j}}-4\hat{\text{k}}\ \text{and}\ \vec{\text{b}_2}=\ 3\hat{\text{i}}+4\hat{\text{j}}+\hat{\text{k}}$
$\Rightarrow\ (7\hat{\text{i}}+2\hat{\text{j}}-3\hat{\text{k}})=(4\hat{\text{i}}-2\hat{\text{j}}-4\hat{\text{k}})+(3\hat{\text{i}}+4\hat{\text{j}}+\hat{\text{k}})$

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