Let a = i +2 j +3 k , b =2 i +3 j -5 k and c =3 i - j + k be thre… - Vidyadip

MCQ

Let $\overrightarrow{ a }=\hat{ i }+2 \hat{ j }+3 \hat{ k }, \overrightarrow{ b }=2 \hat{ i }+3 \hat{ j }-5 \hat{ k } \quad$ and $\overrightarrow{ c }=3 \hat{ i }-\hat{ j }+\lambda \hat{ k }$ be three vectors. Let $\overrightarrow{ r }$ be a unit vector along $\vec{b}+\vec{c}$. If $\vec{r} \cdot \vec{a}=3,$ then $3 \lambda$ is equal to :

A
27
B
25
C
25
D
21

✓

Answer

$\overrightarrow{r}=k(\overrightarrow{b}+\overrightarrow{c})$
$\overrightarrow{r} \cdot \overrightarrow{a}=3$
$\overrightarrow{r} \cdot \overrightarrow{a}=k(\overrightarrow{b} \cdot \overrightarrow{a}+\overrightarrow{c} \cdot \overrightarrow{a})$
$3=k(2+6-15+3-2+3 \lambda)$
$3=k(-6+3 \lambda)$
$\overrightarrow{r}=k(5 \hat{i}+2 \hat{j}-(5-\lambda) \hat{k})$
$|\overrightarrow{r}|=k \sqrt{25+4+25+\lambda^2-10 \lambda}=1 .....(2)$
$ k =\frac{3}{-6+3 \lambda}=\frac{1}{-2+\lambda} $
put in $(2)$
$4+\lambda^2-4 \lambda=54+\lambda^2-10 \lambda$
$6 \lambda=50$
$3 \lambda=25$

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