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

With respect to the origin

O

, the points

A, B, C, D

have position vectors given by

\vec{O A} = - 3 i + j + 8 k, \vec{O B} = - 10 i + 2 j + 15 k, \vec{O C} = 5 i - 2 j - 2 k, \vec{O D} = i + 6 j + 10 k .

Prove that the lines

A B

and

C D

intersect.

Accepted Answer

Given that:
$\vec{O A} = - 3 i + j + 8 k, \vec{O B} = - 10 i + 2 j + 15 k, \vec{O C} = 5 i - 2 j - 2 k, \vec{O D} = i + 6 j + 10 k .$

We have to prove that the lines $A B$ and $C D$ intersect.

The position vector of $\vec{A B} = \vec{O B} - \vec{O A}$

$\vec{A B} = - 7 i + j + 7 k$

The position vector of $\vec{C D} = \vec{O D} - \vec{O C}$

$\vec{C D} = - 4 i + 8 j + 12 k$

The vector equation of line $A B$ is $\vec{r} = \vec{O A} + λ [\vec{O B} - \vec{O A}]$

So, $r = - 3 i + j + 8 k + λ (- 7 i + j + 7 k)$

The vector equation of line $A B$ is $C D$

So, $r = 5 i - 2 j - 2 k + μ (- 4 i + 8 j + 12 k)$

Now equate the components

$x = - 7 λ - 3 x = - 4 μ + 5 . . . . . (1)$

$y = λ + 1 y = 8 μ - 2 . . . . . (2)$

$z = 7 λ + 8 z = 12 μ - 2 . . . . . (3)$

Attempt to solve these equations to find the value of $λ$ and of $μ$ that should give a consistent point:

From $(2)$ : $λ = 8 μ - 3$

Substitute into $(1)$ : $- 7 (8 μ - 3) - 3 = - 4 μ + 5$

$\Rightarrow - 7 (8 μ - 3) = - 4 μ + 8$

$\Rightarrow - 56 μ + 21 = - 4 μ + 8$

$\Rightarrow μ = \frac{13}{52} = \frac{1}{4}$

Then, $λ = 8 (\frac{1}{4}) - 3 = - 1$

Check:

$λ = - 1$ $μ = \frac{1}{4}$

$x = 4$ $x = 4$

$y = 0$ $y = 0$

$z = 1$ $z = 1$

The value of $λ$ and $μ$ give a consistent point.

So there is a point of intersection and the lines are intersecting.

Hence, Proved

With respect to the origin $O$ , the points $A, B, C, D$ have position vectors given by $\vec{O A} = - 3 i + j + 8 k, \vec{O B} = - 10 i + 2 j + 15 k, \vec{O C} = 5 i - 2 j - 2 k, \vec{O D} = i + 6 j + 10 k .$ Prove that the lines $A B$ and $C D$ intersect.

Important Questions on Vectors

Learn and Prepare for any exam you want !

With respect to the origin O, the points A, B, C, D have position vectors given byOA→=-3i+j+8k,OB→=-10i+2j+15k,OC→=5i-2j-2k,OD→=i+6j+10k. Prove that the lines AB and CD intersect.

Important Questions on Vectors

Learn and Prepare for any exam you want !

With respect to the origin $O$ , the points $A, B, C, D$ have position vectors given by $\vec{O A} = - 3 i + j + 8 k, \vec{O B} = - 10 i + 2 j + 15 k, \vec{O C} = 5 i - 2 j - 2 k, \vec{O D} = i + 6 j + 10 k .$ Prove that the lines $A B$ and $C D$ intersect.