Question

1. The angle between the lines passing through the points (5, 7, 8), (3, 3, 4) and (-1, -2, 1), (1, 2, 5) is
(A) pie/3
(B) pie/2
(C) pie/6
(D) 0​

Answer 1

Step-by-step explanation:

Vector equation of the line passing through $\rm(5,7,8)\:\: \& \:\: (3,3,4)$ is

$\vec{r} = (3 \hat{i} + 3\hat{j} + 4\hat{k}) + \lambda \{ (5 - 3)\hat{i} + (7 - 3)\hat{j} + (8 - 4)\hat{k}\}$

$\implies\vec{r} = (3 \hat{i} + 3\hat{j} + 4\hat{k}) + \lambda ( 2\hat{i} + 4\hat{j} +4\hat{k})$

$\implies\vec{r} = (3 \hat{i} + 3\hat{j} + 4\hat{k}) + 2\lambda ( \hat{i} + 2\hat{j} +2\hat{k})$

$\implies\vec{r} = (3 \hat{i} + 3\hat{j} + 4\hat{k}) + \alpha ( \hat{i} + 2\hat{j} +2\hat{k})$

Here, $\alpha=2\lambda$ is a constant

Vector equation of the line passing through $\rm(-1,-2,1)\:\: \& \:\: (1,2,5)$ is

$\vec{r} = ( - \hat{i} - 2 \hat{j} + \hat{k}) + \mu \{(1 + 1) \hat{i} + (2 + 2) \hat{j} + (5 - 1) \hat{k} \}$

$\implies \vec{r} = ( - \hat{i} - 2 \hat{j} + \hat{k}) + \mu \{2 \hat{i} + 4\hat{j} + 4 \hat{k} \}$

$\implies \vec{r} = ( - \hat{i} - 2 \hat{j} + \hat{k}) + 2 \mu ( \hat{i} + 2\hat{j} + 2 \hat{k})$

$\implies \vec{r} = ( - \hat{i} - 2 \hat{j} + \hat{k}) + \beta ( \hat{i} + 2\hat{j} + 2 \hat{k})$

Here, $\alpha=2\mu$ is a constant,

Since, both the lines are parallel to the same vector $\hat{i}+2\hat{j}+2\hat{k}$

So, angle between the lines is 0°