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 the Cartesian and vector equations of the straight line joining the points:

(2, 0, 4) and (1, - 2, 3)

.

Accepted Answer

We know that, the Cartesian equation of the line which passes through the points $A (x_{1}, y_{1}, z_{1})$ and $B (x_{2}, y_{2}, z_{2})$ is given by $\frac{x - x_{1}}{x_{2} - x_{1}} = \frac{y - y_{1}}{y_{2} - y_{1}} = \frac{z - z_{1}}{z_{2} - z_{1}}$ .

Let $A (2, 0, 4)$ and $B (1, - 2, 3)$

So, the required equation of a Line in Cartesian form is $\frac{x - 2}{1 - 2} = \frac{y - 0}{- 2 - 0} = \frac{z - 4}{3 - 4}$ i.e., $\frac{x - 2}{- 1} = \frac{y - 0}{- 2} = \frac{z - 4}{- 1}$ i.e., $\frac{x - 2}{1} = \frac{y}{2} = \frac{z - 4}{1}$

We also know that, The vector equation of a straight line in space passing through two distinct points $A$ and $B$ is given by $\vec{r} = \vec{a} + t (\vec{b} - \vec{a}), t \in R$ , where $\vec{a}$ and $\vec{b}$ are the position vectors of the points $A$ and $B$ respectively.

So, the position vectors of the points $A (2, 0, 4)$ and $B (1, - 2, 3)$ are $\vec{a} = 2 \hat{i} + 0 \hat{j} + 4 \hat{k} and \vec{b} = \hat{i} - 2 \hat{j} + 3 \hat{k}$

Hence, the vector equation of the line which passes through the points $(2, 0, 4) and (1, - 2, 3)$ is $\vec{r} = 2 \hat{i} + 0 \hat{j} + 4 \hat{k} + t ((\hat{i} - 2 \hat{j} + 3 \hat{k}) - (2 \hat{i} + 0 \hat{j} + 4 \hat{k}))$ i.e., $\vec{r} = 2 \hat{i} + 0 \hat{j} + 4 \hat{k} + t (- \hat{i} - 2 \hat{j} - \hat{k})$ .

Find the Cartesian and vector equations of the straight line joining the points: $(2, 0, 4) and (1, - 2, 3)$ .

Important Questions on Straight Line in Three Dimensions

Learn and Prepare for any exam you want !

Find the Cartesian and vector equations of the straight line joining the points: (2,0,4) and (1,-2,3).

Important Questions on Straight Line in Three Dimensions

Learn and Prepare for any exam you want !

Find the Cartesian and vector equations of the straight line joining the points: $(2, 0, 4) and (1, - 2, 3)$ .