Name: Equation of a Line in 3D (Symmetric Form)
Uploaded: 2019-07-19
Duration: 11 min 8 s
Description: We have seen the equation of a line in 2D. Here, in this video, we will see the equation of a line in 3D and also see why vectors play an important role in 3...

Question

Find the equations to the line of greatest slope through the point $(7, 2, - 1)$ in the plane $x - 2 y + 3 z = 0$ assuming that the axes are so placed that the plane $2 x + 3 y - 4 z = 0$ is horizontal.

Accepted Answer

We have,

${\vec{n}}_{1} = \hat{i} - 2 \hat{j} + 3 \hat{k}$
$2 x + 3 y - 4 z = 0$

Vector along the line of intersection of the planes is $\vec{v} = {\vec{n}}_{1} \times {\vec{n}}_{2}$
$\vec{v} = |\begin{matrix} \hat{i} & \hat{j} & \hat{k} \\ 1 & - 2 & 3 \\ 2 & 3 & - 4 \end{matrix}|$

$\Rightarrow \vec{v} = \hat{i} (8 - 9) - \hat{j} (- 4 - 6) + \hat{k} (3 + 4)$

$\Rightarrow \vec{v} = - \hat{i} + 10 \hat{j} + 7 \hat{k}$

Required line is perpendicular to $\vec{v} & {\vec{n}}_{1}$ , so

$- a + 10 b + 7 c = 0$

$a - 2 b + 3 c = 0$

Therefore,

$\frac{a}{30 + 14} = \frac{- b}{- 3 - 7} = \frac{c}{2 - 10}$

$\Rightarrow \frac{a}{22} = \frac{b}{5} = \frac{c}{- 4}$

$∴$ Equation of line the greatest slope is

$\frac{x - 7}{22} = \frac{y - 2}{5} = \frac{z + 1}{- 4}$

Find the equations to the line of greatest slope through the point $(7, 2, - 1)$ in the plane $x - 2 y + 3 z = 0$ assuming that the axes are so placed that the plane $2 x + 3 y - 4 z = 0$ is horizontal.

Important Questions on Three Dimensional Geometry

Learn and Prepare for any exam you want !

Find the equations to the line of greatest slope through the point (7,2,-1) in the plane x-2y+3 z=0 assuming that the axes are so placed that the plane 2x+3y-4z=0 is horizontal.

Important Questions on Three Dimensional Geometry

Learn and Prepare for any exam you want !

Find the equations to the line of greatest slope through the point $(7, 2, - 1)$ in the plane $x - 2 y + 3 z = 0$ assuming that the axes are so placed that the plane $2 x + 3 y - 4 z = 0$ is horizontal.