If vectors \( \vec{a} = 3\hat{i} + 2\hat{j} + \lambda\hat{k} \) and \( \vec{b} = 2\hat{i} - 4\hat{j} + 5\hat{k} \), represent the two strips of the Red Cross sign placed outside a doctor's clinic, then the value of \( \lambda \) is :
Show Hint
The dot product is a scalar value; ensure you sum the products of the components correctly.
- \( 1 \)
- \( \frac{5}{2} \)
- \( \frac{2}{5} \)
- \( 0 \)
The Correct Option is C
Solution and Explanation
Concept:
Two non-zero vectors are perpendicular if and only if their dot product is zero: \[ \vec{a}\cdot\vec{b}=0 \]
Step 1: Apply the condition of perpendicularity
Since the vectors represent the perpendicular strips of a Red Cross sign: \[ \vec{a}\cdot\vec{b}=0 \]
Step 2: Calculate the dot product
Given: \[ \vec{a}=3\hat{i}+2\hat{j}+\lambda\hat{k} \] and \[ \vec{b}=2\hat{i}-4\hat{j}+5\hat{k} \] Therefore, \[ (3\hat{i}+2\hat{j}+\lambda\hat{k}) \cdot (2\hat{i}-4\hat{j}+5\hat{k})=0 \] \[ (3)(2)+(2)(-4)+(\lambda)(5)=0 \] \[ 6-8+5\lambda=0 \]
Step 3: Solve for \(\lambda\)
\[ -2+5\lambda=0 \] \[ 5\lambda=2 \] \[ \lambda=\frac{2}{5} \]
Final Answer:
Therefore, \[ \boxed{\lambda=\frac{2}{5}} \] Hence, the correct answer is Option (C).
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