Name: Distance Between Two Parallel Lines in 3D
Uploaded: 2019-07-19
Duration: 6 min 41 s
Description: What do you know about parallel lines in three-dimensional space? Do they stay at a constant distance apart? How do we find that distance? This video answers...

Question

Derive the vector equation of a line passing through a given point and parallel to a given vector and hence obtain the Cartesian equation of the line.

Accepted Answer

Let $\vec{r}$ be any position vector on the line ' $L$ ' and vector a given point on $L$ .

Clearly, $\vec{AP} ∥ \vec{b}$

i.e. $\vec{A P} = λ \vec{b}$ , for some real number $λ$ .

But, $\vec{A P} = \vec{O P} - \vec{O A}$

$\Rightarrow \vec{A P} = \vec{r} - \vec{a}$

$\Rightarrow \vec{r} - \vec{a} = λ \vec{b} - (1)$

Cartesian form:

Let $\vec{r} = x \hat{i} + y \hat{j} + z \hat{k}$

and $\vec{a} = x_{1} \hat{i} + y_{1} \hat{j} + z_{1} \hat{k}$

and $\vec{b} = a \hat{i} + b \hat{j} + c \hat{k}$

But, $\vec{r} - \vec{a} = (x - x_{1}) \hat{i} + (y - y_{1}) \hat{j} + (z - z_{1}) \hat{k}$

$\vec{r} - \overset{⇀}{a} = λ \vec{b}$ [But from $(1)$ ]

$\Rightarrow (x - x_{1}) \hat{i} + (y - y_{1}) \hat{j} + (z - z_{1}) \hat{k} = λ [a \hat{i} + b \hat{j} + c \hat{k}]$

Equating the corresponding components of $\hat{i}, \hat{j}$ and $\hat{k}$ respectively, we get:

$x - x_{1} = a λ, y - y_{1} = b λ, z - z_{1} = c λ$

$\Rightarrow \frac{x - x_{1}}{a} = λ, \frac{y - y_{1}}{b} = λ, \frac{z - z_{1}}{c} = λ$

Thus, we have:

$\frac{x - x_{1}}{a} = \frac{y - y_{1}}{b} = \frac{z - z_{1}}{c}$

Derive the vector equation of a line passing through a given point and parallel to a given vector and hence obtain the Cartesian equation of the line.

Important Questions on Three Dimensional Geometry

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