Question

Line through \((1,2,3)\) parallel to \(\hat{i}+2\hat{j}+3\hat{k}\):

• A. \(\frac{x-1}{1}=\frac{y-2}{2}=\frac{z-3}{3}\)
• B. \(\frac{x}{1}=\frac{y}{2}=\frac{z}{3}\)
• C. \(\frac{x+1}{1}=\frac{y+2}{2}=\frac{z+3}{3}\)
• D. \(\frac{x-1}{2}=\frac{y-2}{3}=\frac{z-3}{1}\)

Hint:

Line through point: \[ (x_1,y_1,z_1) \] parallel to vector: \[ a\hat{i}+b\hat{j}+c\hat{k} \] has equation: \[ \frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c} \]

Answer 1

The Correct Option is A

Concept: Equation of a line passing through point: \[ (x_1,y_1,z_1) \] and parallel to vector: \[ a\hat{i}+b\hat{j}+c\hat{k} \] is: \[ \frac{x-x_1}{a} = \frac{y-y_1}{b} = \frac{z-z_1}{c} \]

Step 1: Identifying point and direction ratios. Given point: \[ (1,2,3) \] Direction vector: \[ \hat{i}+2\hat{j}+3\hat{k} \] Hence direction ratios are: \[ 1,\ 2,\ 3 \]

Step 2: Writing line equation. Using standard symmetric form: \[ \frac{x-1}{1} = \frac{y-2}{2} = \frac{z-3}{3} \] Final Answer: \[ \boxed{(A)\ \frac{x-1}{1}=\frac{y-2}{2}=\frac{z-3}{3}} \]