If a = 2 i - j + k, , , b = i + j - 2 k and c = i + 3 j - ( ^2 + … - Vidyadip

MCQ

If $\vec a = 2\hat i - \hat j + \hat k,\,\,\vec b = \hat i + \hat j - 2\hat k$ and $\vec c = \hat i + 3\hat j - \left( {{\lambda ^2} + 3\lambda } \right)\hat k$ (where $\lambda $ is a constant) and $\vec a $ is perpendicular to $\vec c - \lambda \vec b$, then sum of different values of $\lambda$ is

✓
$-1$
B
$1$
C
$4$
D
$-4$

✓

Answer

Correct option: A.

$-1$

a
$(2 \hat{i}-\hat{j}+\hat{k}) \cdot[(1-\lambda) \hat{i}+(3-\lambda) \hat{j}$

$\left.\quad+\left(2 \lambda-\lambda^{2}-3 \lambda\right) \hat{k}\right]=0$

$\Rightarrow \quad 2-2 \lambda-3+\lambda+2 \lambda-\lambda^{2}-3 \lambda=0$

$\Rightarrow \quad-\lambda^{2}-2 \lambda-1=0$

$\Rightarrow \quad(\lambda+1)^{2}=0$

$\Rightarrow \quad \lambda=-1$

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