Name: The Shortest Distance Between Two Skew Lines
Uploaded: 2019-07-19
Duration: 9 min 21 s
Description: Skew lines are the lines that do not intersect and not parallel. The shortest distance between two skew lines lies along the line, which is perpendicular to...

Question

Find in vector form as well as in cartesian form, the equation of the line passing through the points

A (1, 2, - 1)

and

B (2, 1, 1)

Accepted Answer

We know that the vector equation of a line passing through the points with position vectors $\vec{a}$ and $\vec{b}$ is $\vec{r} = \vec{a} + λ (\vec{b} - \vec{a}),$ where $λ$ is a scalar.

Here, $\vec{a} = \hat{i} + 2 \hat{j} - \hat{k}$

$\vec{b} = 2 \hat{i} + \hat{j} + \hat{k}$

Vector equation of the required line is

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

$\Rightarrow \vec{r} = (\hat{i} + 2 \hat{j} - \hat{k}) + λ (\hat{i} - \hat{j} + 2 \hat{k}) \dots (1)$

Here, $λ$ is a parameter.

Reducing $(1)$ to cartesian form, we get

$x \hat{i} + y \hat{j} + z \hat{k} = (\hat{i} + 2 \hat{j} - \hat{k}) + λ (\hat{i} - \hat{j} + 2 \hat{k})$ [Putting $\vec{r} = x \hat{i} + y \hat{j} + z \hat{k}$ ]

$\Rightarrow x \hat{i} + y \hat{j} + z \hat{k} = (1 + λ) \hat{i} + (2 - λ) \hat{j} + (- 1 + 2 λ) \hat{k}$

Comparing the coefficients of $\hat{i}, \hat{j}$ and $\hat{k},$ we get

$x = 1 + λ, y = 2 - λ, z = - 1 + 2 λ$

$\Rightarrow x - 1 = λ, \frac{y - 2}{- 1} = λ, \frac{z + 1}{2} = λ$

Hence, the Cartesian form is $\frac{x - 1}{1} = \frac{y - 2}{- 1} = \frac{z + 1}{2}$ .

Find in vector form as well as in cartesian form, the equation of the line passing through the points $A (1, 2, - 1)$ and $B (2, 1, 1)$

Important Questions on Straight Line in Space

Learn and Prepare for any exam you want !

Find in vector form as well as in cartesian form, the equation of the line passing through the points A(1,2,-1) and B(2,1,1)

Important Questions on Straight Line in Space

Learn and Prepare for any exam you want !

Find in vector form as well as in cartesian form, the equation of the line passing through the points $A (1, 2, - 1)$ and $B (2, 1, 1)$