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

Three lines, $L_{1}, L_{2}$ and $L_{3},$ have vector equations

$r = 16 i - 4 j - 6 k + λ (- 12 i + 4 j + 3 k)$
$r = 16 i + 28 j + 15 k + μ (8 i + 8 j + 5 k)$
$r = i + 9 j + 3 k + v (4 i - 12 j - 8 k)$

The $3$ points of intersection of these lines form an acute-angled triangle. For this triangle, find the length of each side.

Accepted Answer

Given that: Three lines, $L_{1}, L_{2}$ and $L_{3},$ have vector equations

$r = 16 i - 4 j - 6 k + λ (- 12 i + 4 j + 3 k)$
$r = 16 i + 28 j + 15 k + μ (8 i + 8 j + 5 k)$
$r = i + 9 j + 3 k + v (4 i - 12 j - 8 k)$

We have to find the length of each side.

We have, $r = 16 i - 4 j - 6 k + λ (- 12 i + 4 j + 3 k)$ and $r = 16 i + 28 j + 15 k + μ (8 i + 8 j + 5 k)$

Now equate the components

$x = - 12 λ + 16 x = 8 μ + 16 . . . . . (1)$

$y = 4 λ - 4 y = 8 μ + 28 . . . . . (2)$

$z = 3 λ - 6 z = 5 μ + 15 . . . . . (3)$

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

From $(2)$ : $λ = 2 μ + 8$

Substitute into $(3)$ : $3 (2 μ + 8) - 6 = 5 μ + 15$

$\Rightarrow 3 (2 μ + 8) = 5 μ + 21$

$\Rightarrow μ + 24 = 21$

$\Rightarrow μ = - 3$

Then, $λ = 2 (- 3) + 8 = 2$

Check:

$λ = 2$ $μ = - 3$

$x = - 8$ $x = - 8$

$y = 4$ $y = 4$

$z = 0$ $z = 0$

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

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

Therefore, the position vector of vertex is $- 8 i + 4 j + 0 k$ .

Similarly, the position vector of vertices are $4 i - 3 k, 12 j + 5 k$

The position vectors of sides are $- 12 i + 4 j + 3 k, 8 i + 8 j + 5 k, - 4 i + 12 j + 8 k$

Now, the length of first side is $|- 12 i + 4 j + 3 k| = \sqrt{{(- 12)}^{2} + 4^{2} + 3^{2}} = 13$

The length of second side is $|8 i + 8 j + 5 k| = \sqrt{8^{2} + 8^{2} + 5^{2}} = 3 \sqrt{17}$

The length of third side is $|- 4 i + 12 j + 8 k| = \sqrt{{(- 4)}^{2} + 12^{2} + 8^{2}} = 4 \sqrt{14}$

Hence, find the length of each side are $13, 4 \sqrt{14}, 3 \sqrt{17}$ .

Important Questions on Vectors

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Three lines, L1, L2 and L3, have vector equations r=16i-4j-6k+λ(-12i+4j+3k) r=16i+28j+15k+μ(8i+8j+5k) r=i+9j+3k+v(4i-12j-8k) The 3 points of intersection of these lines form an acute-angled triangle. For this triangle, find the length of each side.

Important Questions on Vectors

Learn and Prepare for any exam you want !