Let a be a vector which is perpendicular to the vector 3 i +1 2 j… - Vidyadip

MCQ

Let $\vec{a}$ be a vector which is perpendicular to the vector $3 \hat{ i }+\frac{1}{2} \hat{ j }+2 \hat{ k }$. If $\overrightarrow{ a } \times(2 \hat{ i }+\hat{ k })=2 \hat{ i }-13 \hat{ j }-4 \hat{ k }$, then the projection of the vector $\vec{a}$ on the vector $2 \hat{ i }+2 \hat{ j }+\hat{ k }$ is

A
$\frac{1}{3}$
B
$1$
✓
$\frac{5}{3}$
D
$\frac{7}{3}$

✓

Answer

Correct option: C.

$\frac{5}{3}$

c
$(\vec{a} \times(2 \hat{i}+\hat{k})) \times\left(3 \hat{i}+\frac{1}{2} \hat{j}+2 \hat{k}\right)$

$=(2 \hat{i}-13 \hat{j}-4 \hat{k}) \times\left(3 \hat{i}+\frac{1}{2} \hat{j}+2 \hat{k}\right)$

$-(6+2) \overrightarrow{ a }=\left|\begin{array}{ccc}\hat{ i } & \hat{ j } & \hat{ k } \\ 2 & -13 & -4 \\ 3 & \frac{1}{2} & 2\end{array}\right|$

$\vec{a}=3 \hat{i}+2 \hat{j}-5 \hat{k}$

Projection of $\vec{a}$ on vector $2 \hat{i}+2 \hat{j}+\hat{k}$ is

$\overrightarrow{ a } \cdot \frac{(2 \hat{ i }+2 \hat{ j }+\hat{ k })}{3}=\frac{5}{3}$

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