The image of the point (1, 0, 3) about the line passing through vector a = 3i + 2j - k

Question

The image of the point (1, 0, 3) about the line passing through \(\vec{a}=3 \hat{i}+2 \hat{j}-\hat{k}\) and whose direction ratios are \(\vec{r}=4 \hat{i}+2 \hat{j}-\hat{k}\) is

(1) \(\left(\frac{-23}{21}, \frac{20}{21}, \frac{-73}{21}\right)\)

(2) \(\left(\frac{1}{21}, \frac{-23}{21}, \frac{-31}{21}\right)\)

(3) \(\left(\frac{1}{21}, \frac{21}{23}, \frac{-30}{21}\right)\)

(4) \(\left(\frac{3}{21}, \frac{7}{21}, \frac{-5}{21}\right)\)

Answer 1

Correct option is: (1) \(\left(\frac{-23}{21}, \frac{20}{21}, \frac{-73}{21}\right)\) 

\(\vec{a}: 3 \hat{i}+2 \hat{j}-\hat{k}\)

Dr: \(4 \hat{i}+2 \hat{j}-\hat{k}\)

L: \(\frac{x-3}{4}=\frac{y-2}{2}=\frac{z+1}{-1}\)   

Any point on line L: \(P^{\prime}(4 \lambda+3,2 \lambda+2,-\lambda-1)\)

\(P P^{\prime} \cdot \vec{n}=0\)

\(\Rightarrow 4(4 \lambda+2)+2(2 \lambda+2)+(-\lambda-4)(-1)=0\)

\(16 \lambda+8+4 \lambda+4 \lambda+4=021 \lambda+16=0\)