Question

A vector perpendicular to the line \( \vec{r} = \hat{i} + \hat{j} - \hat{k} + \lambda (3\hat{i} - \hat{j}) \) is:

• A. \( 5\hat{i} + \hat{j} + 6\hat{k} \)
• B. \( \hat{i} + 3\hat{j} + 5\hat{k} \)
• C. \( 2\hat{i} - 2\hat{j} \)
• D. \( 9\hat{i} - 3\hat{j} \)

Hint:

To check for perpendicularity, always calculate the dot product of the given vectors and ensure it equals zero.

Answer 1

The Correct Option is B

Step 1: Understanding the direction vector of the line. The given line can be expressed as: \[ \vec{r} = \hat{i} + \hat{j} - \hat{k} + \lambda (3\hat{i} - \hat{j}) \] Here, the direction vector of the line is: \[ \vec{d} = 3\hat{i} - \hat{j} \] 

Step 2: Condition for perpendicularity. A vector \( \vec{v} \) is perpendicular to \( \vec{d} \) if their dot product is zero: \[ \vec{v} \cdot \vec{d} = 0 \] Let \( \vec{v} = a\hat{i} + b\hat{j} + c\hat{k} \). The dot product is: \[ \vec{v} \cdot \vec{d} = (a\hat{i} + b\hat{j} + c\hat{k}) \cdot (3\hat{i} - \hat{j}) = 3a - b \] For perpendicularity: \[ 3a - b = 0 \quad \Rightarrow \quad b = 3a \quad \cdots (1) \] 

Step 3: Using the options to find the correct vector. We substitute the options to check which satisfies \( b = 3a \):   For \( \hat{i} + 3\hat{j} + 5\hat{k} \) 
(Option \( B \)): \[ a = 1, \, b = 3, \, c = 5 \] Substituting into \( b = 3a \): \[ b = 3 \times 1 = 3 \quad \text{(True)} \] Therefore, this vector satisfies the condition. 
 Other options do not satisfy \( b = 3a \). 

Conclusion: Thus, the required vector is \( \hat{i} + 3\hat{j} + 5\hat{k} \), which corresponds to option \( \mathbf{(B)} \).