Question

The vector equation of the straight line \(\frac{x-2}{3} = \frac{y+1}{2} = \frac{z-3}{2}\) is:

• A. \((2\hat{i} - \hat{j} - 3\hat{k}) + \mu(3\hat{i} - 2\hat{j} - 2\hat{k}) \)
• B. \((2\hat{i} - \hat{j} + 3\hat{k}) + \mu(3\hat{i} + 2\hat{j} + 2\hat{k}) \)
• C. \((3\hat{i} - 2\hat{j} - 2\hat{k}) + \mu(2\hat{i} - \hat{j} - 3\hat{k}) \)
• D. \((3\hat{i} + 2\hat{j} + 2\hat{k}) + \mu(2\hat{i} - \hat{j} + 3\hat{k}) \)
• E. \((3\hat{i} + 2\hat{j} + 2\hat{k}) + \mu(2\hat{i} + \hat{j} + 3\hat{k}) \)

Hint:

Always extract point by equating numerator = 0.

Answer 1

The Correct Option is B

Concept: From symmetric form: \[ \frac{x-x_1}{l} = \frac{y-y_1}{m} = \frac{z-z_1}{n} \] Vector equation: \[ \vec{r} = \vec{a} + \mu \vec{d} \]

Step 1: Identify point.
\[ x-2=0,\; y+1=0,\; z-3=0 \Rightarrow (2,-1,3) \]

Step 2: Find direction ratios.
\[ (3,2,2) \]

Step 3: Write vector equation.
\[ \vec{r} = (2\hat{i} - \hat{j} + 3\hat{k}) + \mu(3\hat{i} + 2\hat{j} + 2\hat{k}) \]