KCET · Maths · Definite Integration
If \( \vec{a}=2 \hat{i}+\lambda \hat{j}+\hat{k} \) and \( \vec{b}=\hat{i}+2 \hat{j}+3 \hat{k} \) are orthogonal then the value of \( \lambda \) is
- A \( 00 \)
- B \( 11 \)
- C \( \frac{3}{2} \)
- D \( -\frac{5}{2} \)
Answer & Solution
Correct Answer
(D) \( -\frac{5}{2} \)
Step-by-step Solution
Detailed explanation
Given,
\[
\begin{array}{l}
\vec{a}=2 \hat{i}+\lambda \hat{j}+\hat{k} \rightarrow(1) \\
\vec{b}=\hat{i}+2 \hat{j}+3 \hat{k} \rightarrow(2)
\end{array}
\]
For vectors to be perpendicular,
\[
\begin{array}{l}
\vec{a} \cdot \vec{b} \\
\Rightarrow 2 \cdot 1+2 \cdot \lambda+3 \cdot 1=0 \\
\Rightarrow 5+2 \lambda=0 \\
\Rightarrow \lambda=-\frac{5}{2}
\end{array}
\]
\[
\begin{array}{l}
\vec{a}=2 \hat{i}+\lambda \hat{j}+\hat{k} \rightarrow(1) \\
\vec{b}=\hat{i}+2 \hat{j}+3 \hat{k} \rightarrow(2)
\end{array}
\]
For vectors to be perpendicular,
\[
\begin{array}{l}
\vec{a} \cdot \vec{b} \\
\Rightarrow 2 \cdot 1+2 \cdot \lambda+3 \cdot 1=0 \\
\Rightarrow 5+2 \lambda=0 \\
\Rightarrow \lambda=-\frac{5}{2}
\end{array}
\]
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