Question

If the vector \( 2\hat{i} + 3\hat{j} + 4\hat{k} \) is perpendicular to the vector \( 5\hat{i} - \lambda\hat{j} + 2\hat{k} \), the value of \( \lambda \) is:

• A. 3
• B. 0
• C. 4
• D. 6

Hint:

The condition for perpendicular vectors is a dot product of zero. The condition for parallel vectors is that their cross product is zero, or one is a scalar multiple of the other. Remember these fundamental vector properties.

Answer 1

The Correct Option is D

Step 1: Understanding the Concept:
Two non-zero vectors are perpendicular if and only if their dot product (scalar product) is zero.
Step 2: Key Formula or Approach:
Let \( \vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k} \) and \( \vec{b} = b_1\hat{i} + b_2\hat{j} + b_3\hat{k} \).
If \( \vec{a} \perp \vec{b} \), then \( \vec{a} \cdot \vec{b} = 0 \).
The dot product is calculated as: \( a_1b_1 + a_2b_2 + a_3b_3 \).
Step 3: Detailed Explanation or Calculation:
Let \( \vec{a} = 2\hat{i} + 3\hat{j} + 4\hat{k} \) and \( \vec{b} = 5\hat{i} - \lambda\hat{j} + 2\hat{k} \).
Since the vectors are perpendicular, their dot product must be zero:
\[ \vec{a} \cdot \vec{b} = 0 \] \[ (2)(5) + (3)(-\lambda) + (4)(2) = 0 \] \[ 10 - 3\lambda + 8 = 0 \] \[ 18 - 3\lambda = 0 \] \[ 18 = 3\lambda \] \[ \lambda = \frac{18}{3} = 6 \] Step 4: Final Answer:
The value of \( \lambda \) is 6.