JEE Mains · Maths · STD 12 - 10. vector algebra
If the vectors \(\vec{a}=\lambda \hat{i}+\mu \hat{j}+4 \hat{k}, \vec{b}=2 \hat{i}+4 \hat{j}-2 \hat{k}\) and \(\vec{c}=2 \hat{i}+3 \hat{j}+\hat{k}\) are coplanar and the projection of \(\vec{a}\) on the vector \(\vec{b}\) is \(\sqrt{54}\) units, then the sum of all possible values of \(\lambda+\mu\) is equal to
- A \(0\)
- B \(6\)
- C \(24\)
- D \(18\)
Answer & Solution
Correct Answer
(C) \(24\)
Step-by-step Solution
Detailed explanation
\(\left|\begin{array}{ccc}\lambda & \mu & 4 \\ -2 & 4 & -2 \\ 2 & 3 & 1\end{array}\right|=0\) \(\lambda(10)=\mu(2)+4(-14)=0\) \(10 \lambda-2 \mu=56\) \(5 \lambda-\mu=28\) \(\frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}=\sqrt{54}\) \(\frac{-2 \lambda+4 \mu-8}{\sqrt{24}}=\sqrt{54}\)…
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