Worked examples — Angle between two lines

This page is the case zoo for Angle between two lines. The parent gave you the recipe: $\cos \theta =\frac{|\overrightarrow{b_1}\cdot \overrightarrow{b_2}|}{|\overrightarrow{b_1}||\overrightarrow{b_2}|}.$ Here we hunt down every way this problem can be asked — every sign pattern, every degenerate input, every limiting angle, plus a word problem and an exam twist — and grind each one to a number.

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The scenario matrix

Before any computing, here is the full list of "shapes" this topic can take. Every worked example below is tagged with the cell it kills.

Cell	What makes it tricky	Example
A All-positive ratios	warm-up, no sign traps	Ex 1
B Mixed signs	the modulus actually does something	Ex 2
C Perpendicular ($\theta =90^{\circ }$)	dot product hits exactly $0$	Ex 3
D Parallel / anti-parallel ($\theta =0^{\circ }$)	$-\vec{b}$ is the same line	Ex 4
E Given in symmetric form	must read off directions, ignore the point	Ex 5
F Missing component	solve for a letter using a condition	Ex 6
G Cross-product / $\sin$ route	when $\cos$ is awkward	Ex 7
H Word problem (real-world)	translate objects into direction vectors	Ex 8
I Exam twist (direction cosines + degenerate)	denominator $=1$, spot the zero vector trap	Ex 9

Two quick reminders we lean on constantly:

Recall What is

$\overrightarrow{b_1}\cdot \overrightarrow{b_2}$ again? For $\overrightarrow{b_1}=(a_1,b_1,c_1)$, $\overrightarrow{b_2}=(a_2,b_2,c_2)$: $a_1{a}_2+b_1{b}_2+c_1{c}_2$. See Dot product of vectors.

Recall Why the modulus

$|\cdot |$? A line points both ways, so $\vec{b}$ and $-\vec{b}$ describe the same line. Taking $|\text{numerator}|$ forces the answer to be the acute angle $0^{\circ }\le \theta \le 90^{\circ }$.

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Ex 1 — Cell A: all-positive ratios

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Ex 2 — Cell B: mixed signs (the modulus earns its keep)

This is the case where forgetting the modulus would still work — but see the next example where it does not.

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Ex 3 — Cell B/C: modulus flips an obtuse vector-angle to an acute line-angle

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Ex 4 — Cell D: parallel and anti-parallel ($\theta =0^{\circ }$)

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Ex 5 — Cell E: given in symmetric form (read off the direction, drop the point)

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Ex 6 — Cell F: solve for a missing component

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Ex 7 — Cell G: the $\sin \theta$ (cross-product) route

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Ex 8 — Cell H: real-world word problem

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Ex 9 — Cell I: exam twist (direction cosines + degenerate trap)

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Recap: which cell did each example hit?

Recall Self-test before you close the tab

Which example handled anti-parallel lines? ::: Ex 4 (Cell D) — proportional ratios, $\cos \theta =1$, $\theta =0^{\circ }$. Which one showed the modulus flipping an obtuse vector-angle? ::: Ex 3 (Cell B/C) — $-1\to |-1|$, $135^{\circ }\to 45^{\circ }$. Which used the cross product and why? ::: Ex 7 (Cell G) — $\sin \theta$ form, useful when ${\vec{b}}_1\times {\vec{b}}_2$ is handy. Why is $(0,0,0)$ never a valid direction? ::: Its magnitude is $0$, making $\cos \theta =\frac{0}{0}$ undefined; a line must point somewhere. In symmetric form, what do you read off — point or denominators? ::: The denominators $(a,b,c)$; the point is irrelevant to angle (Ex 5).

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Connections

• Angle between two lines — the parent recipe these examples exercise.
• Dot product of vectors — engine of the $\cos \theta$ route.
• Cross product of vectors — engine of the $\sin \theta$ route (Ex 7).
• Direction cosines and direction ratios — the inputs; Ex 9 uses the unit-vector shortcut.
• Equation of a line in 3D — where Ex 5's directions were read off.
• Angle between two planes · Angle between a line and a plane · Shortest distance between two lines — same direction-vector machinery, next stops.

Concept Map