Find parametric and symmetric equations of a line

e.g. “ Find the vector parametric, scalar parametric and symmetric equations for the line containing the given point and with the given direction ”

Remember: One point, one direction.

See it in a problem3 worked examples

(2, -1, 3)

1
$P$ $P = (2, -1, 3)$
2
$\mathbf d$ $\mathbf d = (5, 1, 3) - (2, -1, 3) = \langle 3, 2, 0\rangle$
3
$\langle x, y, z\rangle = P + t\,\mathbf d$ $\langle x, y, z\rangle = \langle 2, -1, 3\rangle + t\langle 3, 2, 0\rangle$
4
$x$ $x = 2 + 3t$

\langle x, y, z\rangle = \langle 2, -1, 3\rangle + t\langle 3, 2, 0\rangle

Common mistakes3

Dividing by a $0$ in symmetric form. If $d_3 = 0$ , write $\dfrac{x - x_0}{d_1} = \dfrac{y - y_0}{d_2}$ , $z = z_0$ .
Using a 2D normal as the direction. The normal is perpendicular to the line: swap its components and change one sign.
Your answer can differ from the key and still be right. Any point on the line works, and so does any nonzero multiple of the direction.

2

Find the line of intersection of two planes

e.g. “ Find a vector parametric equation for the line of intersection of the given planes ”

1
$\mathbf n_1$ $Parallel normals mean parallel (or identical) planes: no single line of intersection.$
2
$\mathbf d = \mathbf n_1\times\mathbf n_2$ $Divide out any common factor to keep the numbers small.$
3
$0$ $0$
4
$\langle x, y, z\rangle = P + t\,\mathbf d$ $Check by substituting into both plane equations.$

Remember: Cross the normals, then find one point.

See it in a problem3 worked examples

x + y - z = 2

1
$\mathbf n_1$ $\mathbf n_1 = \langle 1, 1, -1\rangle$
2
$\mathbf d = \mathbf n_1\times\mathbf n_2$ $\mathbf n_1\times\mathbf n_2 = \langle 1 - 1,\ -2 - 1,\ -1 - 2\rangle = \langle 0, -3, -3\rangle$
3
$0$ $z = 0$
4
$\langle x, y, z\rangle = P + t\,\mathbf d$ $\langle x, y, z\rangle = \langle 3, -1, 0\rangle + t\langle 0, 1, 1\rangle$

\langle x, y, z\rangle = \langle 3, -1, 0\rangle + t\langle 0, 1, 1\rangle

Common mistakes3

Setting $z = 0$ when the line never reaches $z = 0$ . This can happen when $d_3 = 0$ . Pick another variable.
Planes like $x + y + z = 3$ and $2x + 2y + 2z = 7$ are parallel and do not meet at all.
Your point can differ from the key and still be right: any point on the line works.

3

e.g. “ In each case, determine whether or not the given pair of lines intersect ”

1
$t$
2
$Set the coordinates equal; solve two of the equations$
3
$Check the third equation$ $True: they meet; plug in to get the point. False: they have no common point.$
4
$Name the case: intersecting, parallel or skew$ $No common point? If the directions are multiples, the lines are parallel. Otherwise they are skew Not parallel, and never meeting. This only happens in 3D. .$

Remember: Two parameters, solve two, test the third.

See it in a problem3 worked examples

\langle 1, 2, 0\rangle + t\langle 1, -1, 2\rangle

1
$t$ $(1 + t,\ 2 - t,\ 2t)$
2
$Set the coordinates equal; solve two of the equations$ $x$
3
$Check the third equation$ $z$
4
$Name the case: intersecting, parallel or skew$ $They meet at a single point: intersecting.$

(3, 0, 4)

Common mistakes3

Using the same $t$ for both lines. They can pass through the same point at different parameter values.
Solving two equations and forgetting to test the third: that test is the whole question.
In 3D, lines that do not meet need not be parallel: they can be skew. If all three equations hold for every $t$ , the two lines are the same line.

4

e.g. “ Use a projection to find the distance from the point \dots to the line \dots ”

1
$P$ $ax + by = c$
2
$\mathbf w = Q - P$
3
$\mathbf w - \text{proj}_{\mathbf d}\mathbf w$ $\text{proj}_{\mathbf d}\mathbf w = \dfrac{\mathbf w\cdot\mathbf d}{|\mathbf d|^2}\,\mathbf d$
4
$The distance is the length of what is left$ $\dfrac{|\mathbf w\times\mathbf d|}{|\mathbf d|}$

Remember: Point to point, then strip off the part along the line.

See it in a problem3 worked examples

(4, 2)

1
$P$ $P = (1, 0)$
2
$\mathbf w = Q - P$ $\mathbf w = (4, 2) - (1, 0) = \langle 3, 2\rangle$
3
$\mathbf w - \text{proj}_{\mathbf d}\mathbf w$ $\mathbf w\cdot\mathbf d = 1$
4
$The distance is the length of what is left$ $\tfrac{8}{5}|\langle 2, 1\rangle| = \tfrac{8\sqrt 5}{5}$

\dfrac{8}{\sqrt 5} = \dfrac{8\sqrt 5}{5}

Common mistakes3

Using $Q$ by itself instead of $\mathbf w = Q - P$ .
Dividing by $|\mathbf d|$ in the projection: the vector projection divides by $|\mathbf d|^2$ .
Distance between two parallel lines: pick any point on one and use this recipe with the other.

5

e.g. “ Calculate the distance between the lines \dots and \dots (given in symmetric form) ”

1
$Read a point and a direction from each line$ $\dfrac{x - x_0}{a} = \dfrac{y - y_0}{b} = \dfrac{z - z_0}{c}$
2
$\mathbf n = \mathbf d_1\times\mathbf d_2$ $\mathbf n = \mathbf 0$
3
$\mathbf w = P_2 - P_1$
4
$= \dfrac{|\mathbf w\cdot\mathbf n|}{|\mathbf n|}$ $0$

Remember: Cross the directions, project the gap.

See it in a problem3 worked examples

\dfrac{x - 1}{2} = \dfrac{y}{-1} = z + 1

1
$Read a point and a direction from each line$ $P_1 = (1, 0, -1)$
2
$\mathbf n = \mathbf d_1\times\mathbf d_2$ $\mathbf n = \langle 2 - 1,\ 1 + 4,\ 2 + 1\rangle = \langle 1, 5, 3\rangle$
3
$\mathbf w = P_2 - P_1$ $\mathbf w = \langle -1, 2, 4\rangle$
4
$= \dfrac{|\mathbf w\cdot\mathbf n|}{|\mathbf n|}$ $\mathbf w\cdot\mathbf n = -1 + 10 + 12 = 21$

\dfrac{21}{\sqrt{35}} = \dfrac{3\sqrt{35}}{5}

Common mistakes3

Dividing by $|\mathbf n|^2$ . For a distance, divide by $|\mathbf n|$ only once.
Reading the point from $\dfrac{x + 2}{3}$ as $x_0 = 2$ . It is $x_0 = -2$ .
Dropping the absolute value and reporting a negative distance.