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VECTOR ALGEBRA — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsVECTOR ALGEBRA5 Marks
Question
If $\vec{\text{a}}=\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}}$ and $=\vec{\text{b}}=\hat{\text{j}}-\hat{\text{k}},$ then find a vector $\vec{\text{c}}$ such that $\vec{\text{a}}\times\vec{\text{c}}=\vec{\text{b}}$ and $\vec{\text{a}}\cdot\vec{\text{c}}=3.$
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Answer

Let $\vec{\text{c}}=\text{x}\hat{\text{i}}+\text{y}\hat{\text{j}}+\text{z}\hat{\text{k}}$
Also, $\vec{\text{a}}=\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}}$ and $=\vec{\text{b}}=\hat{\text{j}}-\hat{\text{k}}$
For $\vec{\text{a}}\times\vec{\text{c}}=\vec{\text{b}},$
$\begin{vmatrix}\hat{\text{i}}&\hat{\text{j}}&\hat{\text{k}} \\1&1&1\\\text{x}&\text{y}&\text{z} \end{vmatrix}=\hat{\text{j}}-\hat{\text{k}}$
$\Rightarrow\hat{\text{i}}(\text{z}-\text{y})-\hat{\text{j}}(\text{z}-\text{z})+\hat{\text{k}}(\text{y}-\text{x})=\hat{\text{j}}-\hat{\text{k}}$
$\therefore\text{z}-\text{y}=0\ ...(\text{i})$
$\text{x}-\text{z}=1\ .....(\text{ii})$
$\text{x}-\text{y}=1\ ....(\text{iii})$
Also, $\vec{\text{a}}\cdot\vec{\text{c}}=3$
$\Rightarrow(\hat{\text{i}}+\hat{\text{j}}+\hat{\text{k}})\cdot(\text{x}\hat{\text{i}}+\text{y}\hat{\text{j}}+\text{z}\hat{\text{k}})=3$
$\Rightarrow\text{x}+\text{y}+\text{z}=3\ .....(\text{iv})$
On solving equations (ii) and (iii), we get
$\Rightarrow2\text{x}-\text{y}-\text{z}=2\ .....(\text{v})$
On solving equations (iv) and (v), we get
$\text{x}=\frac{5}{3}$
$\therefore\text{y}=\frac{5}{3}-1=\frac{2}{3}$ and $\text{z}=\frac{2}{3}$
Now, $\vec{\text{c}}=\frac{5}{3}\hat{\text{i}}+\frac{2}{3}\hat{\text{j}}+\frac{2}{3}\hat{\text{k}}$
$=\frac{1}{3}(5\hat{\text{i}}+2\hat{\text{j}}+2\hat{\text{k}})$

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