## IB Maths SL Vectors, Lines

Algebra, Functions and equations, Circular functions and trigonometry, Vectors, Statistics and probability, Calculus. IB Maths SL Revision Notes

### IB Maths SL Vectors, Lines

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

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

$L_{1}: \frac{x-3}{1}= \frac{1-y}{2}= \frac{z+1}{3}$

and $L_{2}:$

$x=4+2 \mu$
$y=-3-4 \mu$
$z=5+ 6 \mu$

Thanks
Jack

Posts: 0
Joined: Mon Jan 28, 2013 8:08 pm

### Re: IB Maths SL Vectors, Lines

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

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

$L_{1}: \frac{x-3}{1}= \frac{y-1}{-2}= \frac{z+1}{3}$

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

$\begin{pmatrix} 2 \\ -4 \\ 6 \end{pmatrix}=2 \begin{pmatrix} 1 \\ -2 \\ 3 \end{pmatrix}$

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

For example the point
$\begin{pmatrix} 6 \\ -7 \\ 11 \end{pmatrix}$ lies on $L_{2}$ for

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

Set
$6=3+ \lambda$
$-7=1-2 \lambda$
$11=-1+ 3 \lambda$

So there is not common solution for $\lambda$

Therefore the lines $L_{1}\and\ L_{2}$ are parallel.

Hope these help!
miranda

Posts: 268
Joined: Mon Jan 28, 2013 8:03 pm