Correct Answer - Option 3 : 13/4
Concept:
If \(\rm \vec{a} \ and \ \vec b\) are two vectors then \(\rm \vec{a}.\vec{b}=|\vec{a}||\vec{b}|cos\theta\)
Note: If vectors \(\rm \vec{a} \ and \ \vec b\) are perpendicular to each other then \(\rm \vec{a}.\vec{b}=0\)
Calculation:
Given: \(\rm \vec{a}= 2\hat i + \hat j +3\hat k, \rm \vec{b}= \hat i + \hat j- 2\hat k \ and \ \rm \vec{c}= 2\hat i + 3\hat k \)
\(\Rightarrow \rm \vec{a}+λ\vec{b}=( 2\hat i+\hat j +3\hat k ) +λ(\hat i+\hat j-2\hat k)\)
⇒ \(\rm \vec{a}+λ\vec{b}=( 2+λ)\vec i+(1+λ)\vec j +(3-2λ)\vec k \)
Now, \(\rm \vec{a}+λ\vec{b}\) and \(\rm \vec{c}\) are perpendicular
\(\Rightarrow (\rm \vec{a}+λ\vec{b}) \cdot \vec c = 0\)
⇒ \(\rm [( 2+λ)\hat i+(1+λ)\hat j +(3-2λ)\hat k].(2\hat i+3\hat k)=0 \)
⇒ 2(2 + λ) + 3(3 - 2λ) = 0
⇒ 4 + 2λ + 9 - 6λ = 0
⇒ 13 - 4λ = 0
⇒ λ = 13/4
Hence, option 3 is correct.
