Question

For which value of \(\lambda\) is the vector \(2\hat{i}+\hat{j}-\lambda\hat{k}\) perpendicular to the vector \(\hat{i}+4\hat{j}+\hat{k}\)?

• A. 4
• B. 5
• C. 6
• D. -6

Hint:

Perpendicular vectors have zero dot product; solve 6−λ=0.

Answer 1

The Correct Option is C

Step 1: Testing option C directly:
With \(\lambda=6\): vector is \(2\hat i+\hat j-6\hat k\).

Step 2: Dot product check:
\((2)(1)+(1)(4)+(-6)(1)=2+4-6=0\), confirming perpendicularity.

Step 3: Ruling out others:
Any other \(\lambda\) leaves \(6-\lambda\ne 0\), so the dot product is non-zero and the vectors are not perpendicular.

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