## Coincident Lines

Discussions for the Core part of the syllabus. Algebra, Functions and equations, Circular functions and trigonometry, Vectors, Statistics and probability, Calculus. IB Maths HL Revision Notes

### Coincident Lines

Vectors, Coincident - IB Maths HL

How can we determine if the following lines intersect in one point, are parallel and distinct, are coincident, or are skew?

$L_{1}: \frac{x-2}{2}= \frac{1-y}{4}= \frac{z+1}{6}$

and $L_{2}:$

$x=4+6 \mu$
$y=-3-12 \mu$
$z=5+ 18 \mu$
lora

Posts: 0
Joined: Wed Apr 10, 2013 7:36 pm

### Re: Coincident Lines

IB Mathematics HL – Vectors, Parallel, coincident or skew lines

We first rewrite the Cartesian equation for $L_{1}:$

$L_{1}: \frac{x-2}{2}= \frac{y-1}{-4}= \frac{z-(-1)}{3}$

We observe that the lines $L_{1}, L_{2}$ have parallel direction vectors since

$\begin{pmatrix} 6 \\ -12 \\ 18 \end{pmatrix}=3 \begin{pmatrix} 2 \\ 4 \\ 6 \end{pmatrix}$

Thus the lines $L_{1}, L_{2}$ are either parallel or coincident.

For example the point
$\begin{pmatrix} 4 \\ -3 \\ 5 \end{pmatrix}$ lies on $L_{2}$ for

$\mu =0$ . Then we check whether it is also on $L_{1}$

Set $\begin{pmatrix} x \\ y \\ z \end{pmatrix} =\begin{pmatrix} 4 \\ -3 \\ 5 \end{pmatrix}$
into the equation of $L_{1}$

and we have

$\frac{4-2}{2}= \frac{-3-1}{-4}= \frac{5-(-1)}{3}=1$

Therefore the point $\begin{pmatrix} 4 \\ -3 \\ 5 \end{pmatrix}$ lies on $L_{1}$. As they share a common point, $L_{1}\and\ L_{2}$ cannot be parallel, thus they are coincident.

Hope these help!
lily

Posts: 0
Joined: Mon Jan 28, 2013 8:04 pm

### Re: Coincident Lines

Thanks Lily!!
lora

Posts: 0
Joined: Wed Apr 10, 2013 7:36 pm