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What is the value of $\lambda$ for which the vectors
$3 \hat{i}+4 \hat{j}-\hat{k}$ and $-2 \hat{i}+\lambda \hat{j}+10 \hat{k}$ are perpendicular?
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$3 \hat{i}+4 \hat{j}-\hat{k}$ and $-2 \hat{i}+\lambda \hat{j}+10 \hat{k}$ are perpendicular?
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Verified Answer
The correct answer is:
4
Given vectors are $3 \hat{i}+4 \hat{j}-\hat{k}$ and $-2 \hat{i}+\lambda \hat{j}+10 \hat{k}$.
If these are perpendicular, dot product is $0 .$
$(3 \hat{i}+4 \hat{j}-\hat{k}) \cdot(-2 \hat{i}+\lambda \hat{j}+10 \hat{k})=0$
$\Rightarrow(3)(-2)+(4)(\lambda)+(-10)=0$
$\Rightarrow-6+4 \lambda-10=0$
$\Rightarrow 4 \lambda=16$
$\Rightarrow \lambda=4$
If these are perpendicular, dot product is $0 .$
$(3 \hat{i}+4 \hat{j}-\hat{k}) \cdot(-2 \hat{i}+\lambda \hat{j}+10 \hat{k})=0$
$\Rightarrow(3)(-2)+(4)(\lambda)+(-10)=0$
$\Rightarrow-6+4 \lambda-10=0$
$\Rightarrow 4 \lambda=16$
$\Rightarrow \lambda=4$
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