If a = 2 i + 3 j+ k , b = i - j+ k , c = i + j+ k and let d be su… - Vidyadip

MCQ

If $\vec a = 2 \hat i + 3 \hat j+ \hat k ,\vec b = \hat i - \hat j+ \hat k , \vec c = \hat i + \hat j+ \hat k$ and let $\vec d$ be such that $\vec a \times \vec b = \vec d \times \vec b, \vec d. \vec c = 8$, then value of $ \vec d. \vec b$ is -

✓
$6$
B
$-6$
C
$3$
D
$-3$

✓

Answer

Correct option: A.

$6$

a
$(\vec{a}-\vec{d}) \times \vec{b}=0 \Rightarrow \vec{a}=\vec{d}+\lambda \vec{b}$

Dot with $\overrightarrow{\mathrm{c}}$

$2+3+1=8+\lambda(1-1+1)$

$\Rightarrow \lambda=-2$

$\therefore \overrightarrow{\mathrm{d}}=\overrightarrow{\mathrm{a}}+2 \overrightarrow{\mathrm{b}}$

$=4 \hat{i}+\hat{j}+3 \hat{k}$

$\therefore \overrightarrow{\mathrm{a}} \cdot \overrightarrow{\mathrm{d}}=4-1+3=6$

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