Name: Equation of Plane Making Intercepts on Axes
Uploaded: 2019-07-19
Duration: 3 min 49 s
Description: In 2-D geometry, you have studied the Intercept form of the Equation of the straight line. Could you think of the Intercept form of the equation of a plane?...

Question

Derive the equation of the line in space passing through two given points, both in vector and Cartesian form.

Accepted Answer

Let $\vec{a}$ and $\vec{b}$ be the position vectors of two points $A (x_{1}, y_{1}, z_{1})$ and $B (x_{2}, y_{2}, z_{2})$ respectively that are lying on a line.

Let $\vec{r}$ be any position vector of any arbitrary point $P (x, y, z)$ , then $P$ is a point on the line if and only if $\vec{A P} = \vec{O P} - \vec{O A} = \vec{r} - \vec{a}$ and $\vec{A B} = \vec{O B} - \vec{O A} = \vec{b} - \vec{a}$ are collinear vectors.

Therefore, $P$ is on the line if and only if $\vec{A P} = λ \vec{A B}$

$\Rightarrow \vec{r} - \vec{a} = λ (\vec{b} - \vec{a})$

$\Rightarrow \vec{r} = \vec{a} + λ (\vec{b} - \vec{a})$ , where $λ \in R$

Derivation of Cartesian form from vector form:

Consider $\vec{r} = x \hat{ı} + y \hat{ȷ} + z \hat{k}, \vec{a} = x_{1} \hat{ı} + y_{1} \hat{ȷ} + z_{1} \hat{k} and \vec{b} = (x_{2} \hat{ı} + y_{2} \hat{ȷ} + z_{2} \hat{k})$

Substituting the values of $\vec{r} = x \hat{ı} + y \hat{ȷ} + z \hat{k}, \vec{a} = x_{1} \hat{i} + y_{1} \hat{ȷ} + z_{1} \hat{k} and \vec{b} = (x_{2} \hat{ı} + y_{2} \hat{ȷ} + z_{2} \hat{k})$ in equation $\vec{r} = \vec{a} + λ (\vec{b} - \vec{a})$ , then we get:

$x \hat{ı} + y \hat{ȷ} + z \hat{k} = x_{1} \hat{ı} + y_{1} \hat{ȷ} + z_{1} \hat{k} + λ ((x_{2} - x_{1}) \hat{ı} + (y_{2} - y_{1}) \hat{ȷ} + (z_{2} - z_{1}) \hat{k})$

Equating the coefficients of $\hat{i}, \hat{ȷ}, \hat{k}$ , we get:

$x = x_{1} + λ (x_{2} - x_{1}); y = y_{1} + λ (y_{2} - y_{1}); z = z_{1} + λ (z_{2} - z_{1})$

On eliminating $λ$ , then we get:

$\frac{(x - x_{1})}{(x_{2} - x_{1})} = \frac{(y - y_{1})}{(y_{2} - y_{1})} = \frac{(z - z_{1})}{(z_{2} - z_{1})}$ , which is required Cartesian equation.

Derive the equation of the line in space passing through two given points, both in vector and Cartesian form.

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