Chapter 11 · Three Dimensional Geometry

Exercise 11.2 · Q3: Line through $(4, 7, 8)$ and $(2, 3, 4)$ is parallel to the line through $(-1, -2, 1)$ and $(1, 2, 5)$

Exercise 11.2Parallel lines: proportional direction ratios · Section 11.4

The question

Show that the line through the points $(4, 7, 8)$ ⁠, $(2, 3, 4)$ is parallel to the line through the points $(-1, -2, 1)$ ⁠, $(1, 2, 5)$ ⁠.

Exam answer

Let $A(4, 7, 8)$ ⁠, $B(2, 3, 4)$ ⁠, $C(-1, -2, 1)$ and $D(1, 2, 5)$ ⁠.

Direction ratios of $AB$ ⁠:

\begin{aligned} a_1, b_1, c_1 &= 2 - 4,\ 3 - 7,\ 4 - 8 \\ &= -2,\ -4,\ -4 \end{aligned}

Direction ratios of $CD$ ⁠:

\begin{aligned} a_2, b_2, c_2 &= 1 + 1,\ 2 + 2,\ 5 - 1 \\ &= 2,\ 4,\ 4 \end{aligned}

Two lines are parallel if $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$ ⁠.

\frac{a_1}{a_2} = \frac{-2}{2} = -1,\ \ \frac{b_1}{b_2} = \frac{-4}{4} = -1

\frac{c_1}{c_2} = \frac{-4}{4} = -1

So $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$ ⁠: the direction ratios are proportional.

$\therefore$ $AB$ is parallel to $CD$ ⁠. Hence proved.

Want the reason behind every step, the basics and the checks?

The one idea this problem is built on

Parallel lines have the same direction, so their direction ratios are proportional: $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$ ⁠. Get each line's direction ratios by subtracting its two points, then compare the three ratios.

Given, to find, and the plan

Given: The line $AB$ through $A(4, 7, 8)$ ⁠, $B(2, 3, 4)$ and the line $CD$ through $C(-1, -2, 1)$ ⁠, $D(1, 2, 5)$ ⁠.
To show: $AB \parallel CD$ ⁠.
Plan: (1) Direction ratios of $AB$ ⁠. (2) Direction ratios of $CD$ ⁠. (3) Show they are proportional. (4) A closer look: the lines are two different lines.

Basics you need

1Direction ratios of the line through two points

$x_2 - x_1, y_2 - y_1, z_2 - z_1$ (Note in Section 11.2.1). Lesson: direction cosines and direction ratios

2Parallel lines

Lines with direction ratios $a_1, b_1, c_1$ and $a_2, b_2, c_2$ are parallel when $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$ (Section 11.4, condition (ii)). Lesson: the angle between two lines

3Proportional sets

Two sets are proportional when one is a fixed multiple of the other, the same multiple for every entry. Basics: direction cosines

Solution

Direction ratios of $AB$
$\begin{aligned} a_1, b_1, c_1 &= 2 - 4,\ 3 - 7,\ 4 - 8 \\ &= -2,\ -4,\ -4. \end{aligned}$
Second point minus first point, coordinate by coordinate: the components of $\overrightarrow{AB}$ ⁠. All three are negative because every coordinate of $B$ is smaller than the matching coordinate of $A$ ⁠.

Direction ratios from two points (Section 11.2.1). Lesson: the line through two points
Direction ratios of $CD$
$\begin{aligned} a_2, b_2, c_2 &= 1 - (-1),\ 2 - (-2),\ 5 - 1 \\ &= 2,\ 4,\ 4. \end{aligned}$
The same rule with $C$ first and $D$ second. Subtracting a negative adds: $1 - (-1) = 2$ and $2 - (-2) = 4$ ⁠.

Direction ratios from two points. Basics: points in space
Show that they are proportional
$\begin{aligned} \frac{a_1}{a_2} &= \frac{-2}{2} = -1, \\ \frac{b_1}{b_2} &= \frac{-4}{4} = -1, \\ \frac{c_1}{c_2} &= \frac{-4}{4} = -1. \end{aligned}$
The three ratios are equal, so $-2, -4, -4 = -1\,(2, 4, 4)$ ⁠, and $AB \parallel CD$ ⁠.

Oblique projection, not to scale. The arrow $A \to B$ (direction ratios $-2, -4, -4$ ⁠) and the arrow $C \to D$ (direction ratios $2, 4, 4$ ⁠) point in opposite directions along two parallel lines. A parallel projection keeps parallel lines parallel, so the picture shows it truly.

The factor is $-1$ ⁠: the arrow from $A$ to $B$ points exactly opposite to the arrow from $C$ to $D$ ⁠. Opposite directions are still parallel, because a line has no preferred direction; $\overrightarrow{BA}$ ⁠, with $2, 4, 4$ ⁠, points the same way as $\overrightarrow{CD}$ ⁠. The same factor must work for all three coordinates.

Parallel lines: $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$ (Section 11.4). Exercise 11.2 player
A closer look: two different lines

Points of $AB$ are $(4 - 2t, 7 - 4t, 8 - 4t)$ ⁠. For $x = -1$ we need $t = \frac52$ ⁠, and then $y = 7 - 10 = -3 \ne -2$ ⁠. So $C(-1, -2, 1)$ is not on the line $AB$ ⁠.

Proportional direction ratios show the directions agree; they would also hold if the two "lines" were the same line. Since $C$ is not on $AB$ ⁠, the lines $AB$ and $CD$ are two distinct parallel lines. The question only asks for parallel, so this step is extra, but it shows the situation fully.

A line is a point plus a direction. Basics: a line is a point plus a direction

Result

The direction ratios $-2, -4, -4$ and $2, 4, 4$ are proportional (factor $-1$ ⁠), so the line through $(4, 7, 8)$ and $(2, 3, 4)$ is parallel to the line through $(-1, -2, 1)$ and $(1, 2, 5)$ ⁠; the two lines are distinct.

(The answer key prints no answer for this "show that" question.)

Check it with numbers

Both arrows have length $\sqrt{4 + 16 + 16} = 6$ ⁠, so $\cos\theta = \frac{|-4 - 16 - 16|}{6 \times 6} = \frac{36}{36} = 1$ ⁠, giving $\theta = 0^\circ$ ⁠: parallel.

In fact $\overrightarrow{AB} = \overrightarrow{DC} = -2\hat i - 4\hat j - 4\hat k$ ⁠, so $ABCD$ is a parallelogram.

Common mistakes

Expecting equal direction ratios. Parallel needs proportional ratios; here they differ by the factor $-1$ ⁠.
Sign slips such as $1 - (-1) = 0$ ⁠. It is $1 + 1 = 2$ ⁠.
Comparing only two ratios. All three must give the same factor.

What to take away

Parallel lines: $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$ ⁠.
A negative factor means opposite arrows, still parallel lines.
To see whether parallel lines are different, test one point of the second line on the first.

Every fact used, and where it comes from

Fact	Used in	Source
Direction ratios of $PQ$ ⁠: $x_2 - x_1, y_2 - y_1, z_2 - z_1$	Steps 1, 2	Note in Section 11.2.1 (Lesson)
Parallel lines: $\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$	Step 3	Section 11.4, condition (ii) (Lesson)
Points of a line: $\vec r = \vec a + \lambda\vec b$	Step 4	Section 11.3.1 (Basics)

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Exercise 11.2 · Q3: Line through (4,7,8)(4, 7, 8)(4,7,8) and (2,3,4)(2, 3, 4)(2,3,4) is parallel to the line through (−1,−2,1)(-1, -2, 1)(−1,−2,1) and (1,2,5)(1, 2, 5)(1,2,5)

Basics you need

Solution

Direction ratios of ABABAB

Direction ratios of CDCDCD

Show that they are proportional

A closer look: two different lines

Every fact used, and where it comes from

Exercise 11.2 · Q3: Line through $(4, 7, 8)$ and $(2, 3, 4)$ is parallel to the line through $(-1, -2, 1)$ and $(1, 2, 5)$

Direction ratios of $AB$

Direction ratios of $CD$