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If the projection of the vector \hat{i}+2\hat{j}+\hat{k} on the sum of the two vectors 2\hat{i}+4\hat{j}-5\hat{k} and -\lambda \hat{i}+2\hat{j}+3\hat{k} is 1, then \lambda is equal to ________.
 

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best_answer

Sum of vectors ( \vec{b} ) is
\left ( 2-\lambda \right )i+6j-2k

Projection of \vec{a}= i+2j+k on \vec{b} is
\Rightarrow \frac{\vec{a}\cdot \vec{b}}{\left | \vec{b} \right |}= 1
\Rightarrow \left ( 2-\lambda \right )+12-2= \sqrt{\left ( 2-\lambda \right )^{2}+36+4}
\Rightarrow \left ( 12-\lambda \right )^{2}= \left ( 2-\lambda \right )^{2}+40
\Rightarrow \lambda ^{2}+144-24\lambda = \lambda ^{2}+4-4\lambda +40
\Rightarrow \lambda = 5

Posted by

Kuldeep Maurya

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