Vector and Cartesian equations of a lineEdexcel A-Level Further Maths: Revision notes

A line in 3D is fixed by a point $A$ with position vector $\mathbf a$ and a direction vector $\mathbf b$: $\mathbf r=\mathbf a+\lambda\mathbf b.$ Each value of the parameter $\lambda$ gives one point on the line. For the line through points $A$ and $B$, take the direction as $\overrightarrow{AB}$, the position vector of $B$ minus the position vector of $A$. For $A(2,-1,4)$ and $B(5,1,0)$ the direction is $(3,2,-4)$, so $\mathbf r=(2,-1,4)+\lambda(3,2,-4)$. A line has many correct equations: any point on it and any non-zero multiple of the direction vector.

Eliminating $\lambda$ from $x=a_1+\lambda b_1$, $y=a_2+\lambda b_2$ and $z=a_3+\lambda b_3$ gives $\frac{x-a_1}{b_1}=\frac{y-a_2}{b_2}=\frac{z-a_3}{b_3}.$ Example: $\mathbf r=(2,-1,4)+\lambda(3,2,-4)$ becomes $\frac{x-2}{3}=\frac{y+1}{2}=\frac{z-4}{-4}$. To go back, put each fraction equal to $\lambda$ and read off the point from the numerators and the direction from the denominators. If a denominator is zero, that coordinate is constant: for direction $(2,0,1)$ through $(1,3,5)$ write $\frac{x-1}{2}=\frac{z-5}{1},\;y=3$.

Write the general point $(a_1+\lambda b_1,\,a_2+\lambda b_2,\,a_3+\lambda b_3)$. Use one coordinate to find $\lambda$, then check that all three coordinates agree. Example: is $(11,5,-12)$ on $\mathbf r=(2,-1,4)+\lambda(3,2,-4)$? $x$ gives $\lambda=3$, $y$ gives $-1+6=5$ ✓, but $z=4-12=-8\ne-12$. So no.

For $\mathbf r_1=\mathbf a_1+\lambda\mathbf b_1$ and $\mathbf r_2=\mathbf a_2+\mu\mathbf b_2$, use different parameters. Equate the three components to get three equations in $\lambda$ and $\mu$. Solve two of them, then substitute into the third. If it holds the lines intersect, and substituting back gives the point of intersection. If it fails there is no intersection. Example: $l_4:\,(3,1,-2)+\lambda(1,2,-1)$ and $l_5:\,(3,6,-7)+\mu(2,-1,3)$. From $x$: $\lambda=2\mu$. From $y$: $1+4\mu=6-\mu$, so $\mu=1$, $\lambda=2$. Check $z$: $-2-2=-4$ and $-7+3=-4$ ✓. The lines meet at $(5,5,-4)$.

Two lines are parallel if their direction vectors are multiples of each other. They are the same line if a point of one lies on the other, otherwise they are distinct and never meet. If the directions are not parallel, the lines may still not meet. In 3D, two lines that are neither parallel nor intersecting are called skew. Example: $l_4$ with direction $(1,2,-1)$ and $l_6$ with direction $(0,1,1)$ are not parallel; equating components gives $\lambda=-2$, $\nu=-7$, but the $z$ equation then fails, so they are skew. In 2D, non-parallel lines always meet, but in 3D they need not.