Question:
If vectors \(\vec{a} = 3\hat{i} - 2\hat{j} + 4\hat{k}\) and \(\vec{b} = \lambda\hat{i} + \hat{j}\) are perpendicular to each other, then the value of \(\lambda\) is :
If vectors \(\vec{a} = 3\hat{i} - 2\hat{j} + 4\hat{k}\) and \(\vec{b} = \lambda\hat{i} + \hat{j}\) are perpendicular to each other, then the value of \(\lambda\) is :
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Set \(\vec a\cdot\vec b = 3\lambda - 2 = 0\).
Updated On: Oct 22, 2026
- \(3\)
- \(-3\)
- \(\frac{2}{3}\)
- \(-\frac{2}{3}\)
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The Correct Option is C
Solution and Explanation
Step 1: Recall the perpendicular condition:
Two non-zero vectors are perpendicular exactly when their dot product is zero.
\[ \vec{a}\cdot\vec{b} = 0 \]
Step 2: Write the dot product:
Multiply the matching components and add. The vector \(\vec b\) has no \(\hat k\) part, so that component is 0.
\[ \vec{a}\cdot\vec{b} = (3)(\lambda) + (-2)(1) + (4)(0) = 3\lambda - 2 \]
Step 3: Solve for lambda:
\[ 3\lambda - 2 = 0 \Rightarrow \lambda = \frac{2}{3} \]
Step 4: Check the other options:
For \(\lambda = 3\): dot product \(= 9 - 2 = 7 \neq 0\).
For \(\lambda = -3\): dot product \(= -9 - 2 = -11 \neq 0\).
For \(\lambda = -\frac{2}{3}\): dot product \(= -2 - 2 = -4 \neq 0\).
Final Answer:
So \(\lambda = \frac{2}{3}\), which is option (C). \[ \boxed{\lambda = \frac{2}{3}} \]
Two non-zero vectors are perpendicular exactly when their dot product is zero.
\[ \vec{a}\cdot\vec{b} = 0 \]
Step 2: Write the dot product:
Multiply the matching components and add. The vector \(\vec b\) has no \(\hat k\) part, so that component is 0.
\[ \vec{a}\cdot\vec{b} = (3)(\lambda) + (-2)(1) + (4)(0) = 3\lambda - 2 \]
Step 3: Solve for lambda:
\[ 3\lambda - 2 = 0 \Rightarrow \lambda = \frac{2}{3} \]
Step 4: Check the other options:
For \(\lambda = 3\): dot product \(= 9 - 2 = 7 \neq 0\).
For \(\lambda = -3\): dot product \(= -9 - 2 = -11 \neq 0\).
For \(\lambda = -\frac{2}{3}\): dot product \(= -2 - 2 = -4 \neq 0\).
Final Answer:
So \(\lambda = \frac{2}{3}\), which is option (C). \[ \boxed{\lambda = \frac{2}{3}} \]
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