Question

If vectors \(5\hat i-\lambda\hat j+2\hat k\) and \(2\hat i+3\hat j+4\hat k\) are perpendicular to each other, then the value of \(\lambda\) is:

• A. 3
• B. 4
• C. 6
• D. 0

Hint:

Set the dot product of the two vectors to zero and solve for \(\lambda\).

Answer 1

The Correct Option is C

Step 1: Recall the perpendicularity test:
$\vec u\perp\vec v \iff \vec u\cdot\vec v=0$.

Step 2: Multiply component-wise and add:
$5\times2=10$, $(-\lambda)\times3=-3\lambda$, $2\times4=8$. Sum: $10-3\lambda+8=0$.

Step 3: Isolate \(\lambda\):
$3\lambda=18 \Rightarrow \lambda=6$.

Final Answer:
\[ \boxed{\lambda=6} \]