Question

If two vectors \(\overset{⃗}{A} = 4\hat{i}+n\hat{j}+2\hat{k}\) and \(\overset{⃗}{B} = 2\hat{i}+2\hat{j}-\hat{k}\) are mutually perpendicular to each other then value of 'n' is

• A. \(+3\)
• B. \(+1\)
• C. \(-3\)
• D. \(-1\)

Hint:

Perpendicular vectors have zero dot product.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept
Two vectors are perpendicular when the angle between them is 90 degrees, so \(\vec A\cdot\vec B=|A||B|\cos90^\circ=0\).

Step 2: Key Formula or Approach
\[ \vec A\cdot\vec B=A_xB_x+A_yB_y+A_zB_z \]

Step 3: Substitute
\[ (4)(2)+(n)(2)+(2)(-1)=0 \]
\[ 8+2n-2=0\Rightarrow2n=-6 \]
\[ n=-3 \]

Step 4: Check the options
The option \(+3\) comes from a sign slip when moving 6 across. The values \(\pm1\) do not make the sum zero: for \(n=1\) the sum is 8, and for \(n=-1\) it is 4. So only \(n=-3\) works, option (C).

Final Answer:
The dot product 6 + 2n must be zero, so n = -3, option (C). \[ \boxed{-3} \]