Chapter 11 · Three Dimensional Geometry

Miscellaneous · Q1: Angle between lines with direction ratios $a, b, c$ and $b - c, c - a, a - b$

Miscellaneous ExerciseAngle between two lines · Section 11.4

The question

Find the angle between the lines whose direction ratios are $a, b, c$ and $b - c, c - a, a - b$ ⁠.

Exam answer

Here $a_1 = a$ ⁠, $b_1 = b$ ⁠, $c_1 = c$ and

$a_2 = b - c$ ⁠, $b_2 = c - a$ ⁠, $c_2 = a - b$ ⁠.

\begin{aligned} &a_1a_2 + b_1b_2 + c_1c_2 \\ &= a(b - c) + b(c - a) \\ &\quad + c(a - b) \\ &= ab - ac + bc - ab \\ &\quad + ac - bc \\ &= 0 \end{aligned}

If $\theta$ is the angle between the lines, then

\begin{aligned} \cos\theta &= \frac{|a_1a_2 + b_1b_2 + c_1c_2|}{\sqrt{a_1^2 + b_1^2 + c_1^2}} \\ &\quad \times \frac{1}{\sqrt{a_2^2 + b_2^2 + c_2^2}} \\ &= 0 \end{aligned}

(the square roots are not zero, since neither set of direction ratios is $0, 0, 0$ ⁠).

$\therefore\ \theta = 90^\circ$ ⁠, that is, the lines are perpendicular.

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

The one idea this problem is built on

The angle between two lines depends only on their direction ratios, through $a_1a_2 + b_1b_2 + c_1c_2$ ⁠. When this sum is 0, $\cos\theta = 0$ and the lines are at right angles. Here the sum is 0 for every choice of $a$ ⁠, $b$ ⁠, $c$ ⁠: the terms cancel in pairs.

Given, to find, and the plan

Given: Two lines with direction ratios $a, b, c$ and $b - c, c - a, a - b$ ⁠. As direction ratios, $a, b, c$ are not all 0, and $b - c, c - a, a - b$ are not all 0 (so $a, b, c$ are not all equal).
To find: The angle $\theta$ between the lines.
Plan: (1) Name the two sets of direction ratios. (2) Work out $a_1a_2 + b_1b_2 + c_1c_2$ ⁠. (3) Check that the square roots in the formula are not 0. (4) Find $\theta$ ⁠. (5) See why the sum is always 0.

Basics you need

1Angle between two lines

For direction ratios $a_1, b_1, c_1$ and $a_2, b_2, c_2$ ⁠, the acute angle $\theta$ between the lines satisfies (Section 11.4)

\begin{aligned} \cos\theta &= \frac{|a_1a_2 + b_1b_2 + c_1c_2|}{\sqrt{a_1^2 + b_1^2 + c_1^2}} \\ &\quad \times \frac{1}{\sqrt{a_2^2 + b_2^2 + c_2^2}}. \end{aligned}

Lesson: the angle between two lines

2Perpendicular lines

The lines are perpendicular exactly when $a_1a_2 + b_1b_2 + c_1c_2 = 0$ (Section 11.4, condition (i)). This sum is the dot product of the two direction vectors. Basics: the dot product

3Direction ratios are never all zero

Direction ratios are a nonzero multiple of the direction cosines $l, m, n$ ⁠, and $l^2 + m^2 + n^2 = 1$ (Section 11.2). So $0, 0, 0$ is never a set of direction ratios. Basics: direction cosines and ratios

4The cross product (for Step 5)

$\vec u\times\vec v$ is perpendicular to both $\vec u$ and $\vec v$ (Section 10.6.3). Basics: the cross product

Solution

Name the two sets of direction ratios
$\begin{aligned} a_1 &= a, & b_1 &= b, & c_1 &= c, \\ a_2 &= b - c, & b_2 &= c - a, & c_2 &= a - b. \end{aligned}$
The formula pairs the $x$ -ratios, the $y$ -ratios and the $z$ -ratios. In the second set the order matters: $b - c$ goes with $x$ ⁠, $c - a$ with $y$ ⁠, $a - b$ with $z$ ⁠, exactly as listed in the question.

Direction ratios of a line (Section 11.2). Lesson: direction cosines and ratios
The sum of the products is 0
$\begin{aligned} &a_1a_2 + b_1b_2 + c_1c_2 \\ &= a(b - c) + b(c - a) + c(a - b) \\ &= ab - ac + bc - ba + ca - cb \\ &= (ab - ba) + (ca - ac) + (bc - cb) \\ &= 0. \end{aligned}$
Each product appears twice with opposite signs: $ab$ and $-ba$ ⁠, $-ac$ and $+ca$ ⁠, $bc$ and $-cb$ ⁠. Since $ab = ba$ and so on, every pair cancels. Nothing was assumed about $a$ ⁠, $b$ ⁠, $c$ ⁠, so the sum is 0 for all values: it is an identity.

This sum is the dot product of $a\hat i + b\hat j + c\hat k$ with the second direction. Basics: a⃗ · b⃗ = a₁b₁ + a₂b₂ + a₃b₃
The square roots are not zero
$\begin{aligned} &\sqrt{a^2 + b^2 + c^2} > 0, \\ &\sqrt{(b - c)^2 + (c - a)^2 + (a - b)^2} > 0. \end{aligned}$
A sum of squares is 0 only when every term is 0. The first is not 0 because $a, b, c$ are not all 0. The second would be 0 only if $b = c$ ⁠, $c = a$ and $a = b$ ⁠, that is, $a = b = c$ ⁠; then the second set would be $0, 0, 0$ ⁠, which is not a set of direction ratios, so the question rules this case out. So the formula never divides by 0.

Direction ratios are never all zero (Section 11.2). Basics: direction ratios
Find the angle
$\begin{aligned} &\cos\theta \\ &= \frac{|0|}{\sqrt{a^2 + b^2 + c^2}} \\ &\quad \times \frac{1}{\sqrt{(b - c)^2 + (c - a)^2 + (a - b)^2}} \\ &= 0, \end{aligned}$
so $\theta = 90^\circ$ ⁠.

0 divided by a positive number is 0. The formula gives the acute (or right) angle, $0^\circ \le \theta \le 90^\circ$ ⁠, and in that range $\cos\theta = 0$ only at $\theta = 90^\circ$ ⁠. This is condition (i) of Section 11.4: the lines are perpendicular. We never needed the lengths, only the fact that they are not 0.

Section 11.4, the angle formula and condition (i). Lesson: perpendicular lines
Why it always works

The second direction is a cross product:
$\begin{aligned} &(a\hat i + b\hat j + c\hat k)\times(\hat i + \hat j + \hat k) \\ &= \begin{vmatrix} \hat i & \hat j & \hat k \\ a & b & c \\ 1 & 1 & 1 \end{vmatrix} \\ &= (b - c)\hat i - (a - c)\hat j \\ &\quad + (a - b)\hat k \\ &= (b - c)\hat i + (c - a)\hat j \\ &\quad + (a - b)\hat k. \end{aligned}$
The middle term is $-(a \cdot 1 - c \cdot 1) = c - a$ ⁠. So the second line points along $(a\hat i + b\hat j + c\hat k)\times(\hat i + \hat j + \hat k)$ ⁠, and a cross product is perpendicular to each of its two factors, in particular to the first line's direction. The question is built so that the answer is $90^\circ$ whatever $a$ ⁠, $b$ ⁠, $c$ are. It also shows when it fails: if $a = b = c$ ⁠, the two factors are parallel and the cross product is $\vec 0$ ⁠.

The cross product is perpendicular to both factors (Section 10.6.3). Basics: a⃗ × b⃗ stands up from both

Result

$\theta = 90^\circ$ ⁠: the lines are perpendicular, for every $a$ ⁠, $b$ ⁠, $c$ (not all equal).

(Textbook answer: $90^\circ$ ⁠.)

Check it with numbers

$a = 1$ ⁠, $b = 2$ ⁠, $c = 3$ ⁠: the second set is $-1, 2, -1$ ⁠, and $1(-1) + 2(2) + 3(-1) = -1 + 4 - 3 = 0$ ⁠.

$a = 2$ ⁠, $b = -1$ ⁠, $c = 4$ ⁠: the second set is $-5, 2, 3$ ⁠, and $2(-5) + (-1)(2) + 4(3)$ $= -10 - 2 + 12 = 0$ ⁠.

$a = 0.5$ ⁠, $b = 1.7$ ⁠, $c = -2.3$ ⁠: the second set is $4.0, -2.8, -1.2$ ⁠, and $0.5(4.0) + 1.7(-2.8)$ $+\ (-2.3)(-1.2)$ $= 2 - 4.76 + 2.76 = 0$ ⁠.

Common mistakes

Pairing the wrong ratios, for example $a(c - a)$ ⁠. Keep the order: $a$ with $b - c$ ⁠, $b$ with $c - a$ ⁠, $c$ with $a - b$ ⁠.
Sign slips while expanding: $b(c - a) = bc - ab$ and $c(a - b) = ac - bc$ ⁠.
Reading $\cos\theta = 0$ as $\theta = 0^\circ$ ⁠. $\cos 0^\circ = 1$ ⁠; $\cos\theta = 0$ means $\theta = 90^\circ$ ⁠.
Trying special numbers instead of a proof. One example such as $1, 2, 3$ does not show the answer for all $a, b, c$ ⁠; the expansion in Step 2 does.

What to take away

If $a_1a_2 + b_1b_2 + c_1c_2 = 0$ ⁠, the lines are perpendicular; the lengths are not needed (only that they are not 0).
With letters, expand and collect: here the six terms cancel in pairs, for every $a, b, c$ ⁠.
$b - c, c - a, a - b$ is the cross product of $a, b, c$ with $1, 1, 1$ ⁠, so it is always perpendicular to $a, b, c$ ⁠.

Every fact used, and where it comes from

Fact	Used in	Source
Direction ratios; they are never all 0	Steps 1, 3	Section 11.2 (Basics)
$a_1a_2 + b_1b_2 + c_1c_2$ is the dot product of the two directions	Step 2	Section 10.6.1 (Basics)
The angle between two lines from their direction ratios; perpendicular when $a_1a_2 + b_1b_2 + c_1c_2 = 0$	Steps 3, 4	Section 11.4, equation (1) and condition (i) (Lesson)
The cross product as a determinant; it is perpendicular to both factors	Step 5	Section 10.6.3 (Basics)

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Miscellaneous · Q1: Angle between lines with direction ratios a,b,ca, b, ca,b,c and b−c,c−a,a−bb - c, c - a, a - bb−c,c−a,a−b

Basics you need

Solution

Name the two sets of direction ratios

The sum of the products is 0

The square roots are not zero

Find the angle

Why it always works

Every fact used, and where it comes from

Miscellaneous · Q1: Angle between lines with direction ratios $a, b, c$ and $b - c, c - a, a - b$