If a is nonzero vector such that its projections on the vectors 2… - Vidyadip

MCQ

If $\vec{a}$ is nonzero vector such that its projections on the vectors $2 \hat{i}-\hat{j}+2 \hat{k}, \hat{i}+2 \hat{j}-2 \hat{k}$ and $\hat{k}$ are equal, then a unit vector along $\vec{a}$ is:

A
$\frac{1}{\sqrt{155}}(-7 \hat{ i }+9 \hat{ j }+5 \hat{ k })$
B
$\frac{1}{\sqrt{155}}(-7 \hat{ i }+9 \hat{ j }-5 \hat{ k })$
✓
$\frac{1}{\sqrt{155}}(7 \hat{ i }+9 \hat{ j }+5 \hat{ k })$
D
$\frac{1}{\sqrt{155}}(7 \hat{ i }+9 \hat{ j }-5 \hat{ k })$

✓

Answer

Correct option: C.

$\frac{1}{\sqrt{155}}(7 \hat{ i }+9 \hat{ j }+5 \hat{ k })$

(C) $\frac{1}{\sqrt{155}}(7 \hat{ i }+9 \hat{ j }+5 \hat{ k })$
Explanation: Let $\overline{ a }= a _1 \hat{ i }+ a _2 \hat{ j }+ a _3 \hat{ k }$
$a _1^2+ a _2^2+ a _3^2=1$
Let $\overline{ b }=2 \overrightarrow{ i }-\hat{ j }+2 \hat{ k }, \overline{ c }=\overrightarrow{ i }-2 \hat{ j }-2 \hat{ k }$
$\overline{ d }=\hat{ k }$
$\frac{\overline{ a } \cdot \overline{ b }}{| b |}=\frac{\overline{ a } \cdot \overline{ c }}{| c |}=\frac{\overline{ a } \cdot \overline{ d }}{| d |}$
$\frac{2 a _1- a _2+2 a _3}{3}=\frac{ a _1+2 a _2-2 a _3}{3}= a _3$
By solving
$a_1=\frac{7}{\sqrt{155}}, a_2=\frac{9}{\sqrt{155}}, a_3=\frac{5}{\sqrt{155}}$

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