Three Dimensional Geometry · Lesson 3 of 5

Angles and Relationships between Lines

“Compare direction vectors to find angles and recognise parallel or perpendicular directions.”

Learning Objectives

• Explain how to compare directions of lines that do not pass through the same point. • Derive the acute-angle formula from the dot product and connect it with the cross product. • Use direction ratios or direction cosines to test parallelism and perpendicularity. • Read direction vectors accurately from vector, parametric, and Cartesian equations. • Solve for an unknown parameter using a geometric relationship.

Bring the directions together

Two lines in space need not intersect. Even so, we can compare their directions by drawing parallel lines through a common point. Moving a direction without rotating it does not change its angle with another direction. This is why anchor points play no role in an angle calculation.

Choose nonzero direction vectors $u$ and $v$ for the two lines. Their vector angle lies between $0$ and $π$ and depends on how the arrows were chosen. Reversing one arrow replaces this angle by its supplement. The conventional angle between undirected lines is the smaller angle, between $0$ and $π /2$ , so it must give the same answer after such a reversal.

Compare directions at one origin— Parallel translation lets the same angle rule apply to intersecting lines or skew lines.

The dot-product formula

Recall that the dot product measures how strongly one vector points along another. It equals the product of their lengths times the cosine of their vector angle. Taking an absolute value removes the arbitrary sign caused by reversing a direction arrow and selects the smaller line angle.

For $u = (a_{1}, b_{1}, c_{1})$ and $v = (a_{2}, b_{2}, c_{2})$ , their dot product is $a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2}$ . Their lengths are the square roots of the corresponding sums of squares. Substitute these into the dot-product relation to obtain both versions below.

Smaller angle between two linesLaTeX

cos θ 0 = \frac{∣ u \cdot v ∣}{∣ u ∣ ∣ v ∣} = \frac{∣ a _{1} a _{2} + b _{1} b _{2} + c _{1} c _{2} ∣}{a _{1}^{2} + b _{1}^{2} + c _{1}^{2} a _{2}^{2} + b _{2}^{2} + c _{2}^{2}} \leq θ \leq \frac{π}{2}

The vectors are nonzero direction vectors. For perpendicular directions

θ = π /2

; for parallel directions

θ = 0

. In other cases this is the acute angle.

If the directions are given by direction cosines $(l_{1}, m_{1}, n_{1})$ and $(l_{2}, m_{2}, n_{2})$ , both vectors already have length one. The denominator then becomes one, leaving $cos θ = ∣ l_{1} l_{2} + m_{1} m_{2} + n_{1} n_{2} ∣$ . This is a simplification of the same formula, not a separate geometric rule.

Example — An angle from vector equations

Problem
Find the smaller angle between $r = (3, 2, - 4) + λ (1, 2, 2)$ and $r = (5, - 2, 0) + μ (3, 2, 6)$ .

1.Ignore the anchors for this calculation. The direction vectors are $u = (1, 2, 2)$ and $v = (3, 2, 6)$ .
2.Compute $u \cdot v = 3 + 4 + 12 = 19$ . Their lengths are $1 + 4 + 4 = 3$ and $9 + 4 + 36 = 7$ .
3.Therefore $cos θ = ∣19∣/ (3 \cdot 7) = 19/21$ , giving $θ = cos^{- 1} (19/21)$ . No intersection calculation is required.

Example — An angle from Cartesian equations

Problem
Find the smaller angle between $(x + 3) /3 = (y - 1) /5 = (z + 3) /4$ and $(x + 1) /1 = (y - 4) /1 = (z - 5) /2$ .

1.The denominators give directions $(3, 5, 4)$ and $(1, 1, 2)$ . The signs in the numerators affect anchors, not these direction vectors.
2.The dot product is $3 + 5 + 8 = 16$ , and the lengths are $50$ and $6$ .
3.Thus $cos θ = 16/ 300 = 83 /15$ , so $θ = cos^{- 1} (83 /15)$ .

Use directions, not anchor vectors

In $r = a + t b$ , use $b$ in an angle calculation. The vector $a$ points from the origin to one location on the line; it need not point along the line. Also keep the absolute value when finding the smaller angle between undirected lines. A negative dot product alone does not mean this angle is obtuse.

Why the sine formula has squared differences

The cross product provides a complementary way to compare directions. Its magnitude is the area of the parallelogram built on the two direction vectors. That area is the product of their lengths times the sine of the angle, so it vanishes when the directions are parallel.

Starting instead from the cosine formula, use $sin^{2} θ = 1 - cos^{2} θ$ . Expanding the numerator gives the identity $∣ u ∣^{2} ∣ v ∣^{2} - (u \cdot v)^{2} = (a_{1} b_{2} - a_{2} b_{1})^{2} + (b_{1} c_{2} - b_{2} c_{1})^{2} + (c_{1} a_{2} - c_{2} a_{1})^{2}$ . These three squared differences are the squared components of the cross product, in a rearranged order.

Sine of the line angleLaTeX

sin θ = \frac{∣ u \times v ∣}{∣ u ∣ ∣ v ∣} = \frac{( a _{1} b _{2} - a _{2} b _{1} ) ^{2} + ( b _{1} c _{2} - b _{2} c _{1} ) ^{2} + ( c _{1} a _{2} - c _{2} a _{1} ) ^{2}}{a _{1}^{2} + b _{1}^{2} + c _{1}^{2} a _{2}^{2} + b _{2}^{2} + c _{2}^{2}}

All three terms under the numerator square root are added. With direction cosines, the denominator is one. The nonnegative square root agrees with

0 \leq θ \leq π /2

Parallel and perpendicular directions

Two important angle values lead to simpler tests. A right angle makes the dot product zero. A zero line angle makes the cross product zero, which means the direction vectors are proportional. These tests can avoid inverse trigonometric calculations entirely.

Relationship	Vector test	Coordinate interpretation
Perpendicular directions	$u \cdot v = 0$	$a_{1} a_{2} + b_{1} b_{2} + c_{1} c_{2} = 0$
Parallel directions	$u \times v = 0$	$v = k u$ for some nonzero real $k$
Parallel with all relevant denominators nonzero	$v = k u$	$a_{1} / a_{2} = b_{1} / b_{2} = c_{1} / c_{2}$

Use proportionality rather than component division if a component is zero. For instance, $(1, 0, 2)$ and $(3, 0, 6)$ are parallel because the second is three times the first, even though comparing their second components by division would give $0/0$ . Direction tests also say nothing by themselves about whether two lines share a point.

Example — Perpendicular directions from point pairs

Problem
Compare the line through $(1, - 1, 2)$ and $(3, 4, - 2)$ with the line through $(0, 3, 2)$ and $(3, 5, 6)$ .

1.Subtract within each pair to obtain directions $(2, 5, - 4)$ and $(3, 2, 4)$ .
2.Their dot product is $2 \cdot 3 + 5 \cdot 2 + (- 4) \cdot 4 = 6 + 10 - 16 = 0$ .
3.Their directions are perpendicular, so the angle between the lines is $9 0^{\circ}$ . This conclusion compares directions; it does not establish an intersection point.

Example — Parallel directions from point pairs

Problem
Compare the line through $(4, 7, 8)$ and $(2, 3, 4)$ with the line through $(- 1, - 2, 1)$ and $(1, 2, 5)$ .

1.The first direction is $(- 2, - 4, - 4)$ and the second is $(2, 4, 4)$ .
2.The second vector is $- 1$ times the first, so the lines have parallel directions.
3.The negative factor reverses orientation without changing the line angle, which is $0^{\circ}$ . Whether the lines coincide would require a separate shared-point check.

Example — An unknown parameter with unusual numerators

Problem
Find $p$ if $(1 - x) /3 = (7 y - 14) / (2 p) = (z - 3) /2$ and $(7 - 7 x) / (3 p) = (y - 5) /1 = (6 - z) /5$ have perpendicular directions.

1.Name the first common parameter $t$ . Its coordinate equations are $x = 1 - 3 t$ , $y = 2 + (2 p /7) t$ , and $z = 3 + 2 t$ , so use direction $(- 3, 2 p /7, 2)$ .
2.For the second parameter $s$ , obtain $x = 1 - (3 p /7) s$ , $y = 5 + s$ , and $z = 6 - 5 s$ , so use direction $(- 3 p /7, 1, - 5)$ .
3.Perpendicularity requires $(- 3) (- 3 p /7) + (2 p /7) (1) + 2 (- 5) = 0$ . Thus $11 p /7 - 10 = 0$ , giving $p = 70/11$ .
4.This value is nonzero, so the original fractions involving $p$ are defined. Reading the denominators alone would miss both the numerator coefficients and their signs.

A right angle does not prove intersection

In space, a zero dot product shows orthogonal directions. The two lines may still be skew. In this chapter, problems may describe such directions as perpendicular lines; keep the direction conclusion separate from a claim that they meet. For an intersection claim, solve for a common point as well.

Check your understanding

The direction comparison should be unchanged by shifting a line or rescaling its direction vector. Use these invariances as checks on your calculations. In particular, the smaller angle cannot depend on which arrow you chose along an undirected line.

Quick check

What is the smaller angle between lines directed by $(1, 2, 2)$ and $(- 1, - 2, - 2)$ ?

Quick check

Find the smaller angle between directions $(1, 0, 1)$ and $(1, 1, 0)$ .

Quick check

Which pair has parallel directions?

Quick check

The dot product of two nonzero direction vectors is zero. What can be concluded?

Quick check

Why does $∣ u \times v ∣ = 0$ identify parallel directions?

Quick check

A line is shifted to a parallel position. What happens to its angle with a fixed line?

Quick check

Which direction follows from $(2 - x) /4 = (3 y + 6) /9 = (z - 1) /2$ ?

Practice Problems

Show that the three direction-cosine triples $(12, - 3, - 4) /13$ , $(4, 12, 3) /13$ , and $(3, - 4, 12) /13$ are mutually perpendicular.
Find the angle between $r = (2, - 5, 1) + t (3, 2, 6)$ and $r = (7, 0, - 6) + s (1, 2, 2)$ .
Find the angle between $r = (3, 1, - 2) + t (1, - 1, - 2)$ and $r = (2, - 1, - 56) + s (3, - 5, - 4)$ .
Find the angle between $(x - 2) /2 = (y - 1) /5 = (z + 3) / (- 3)$ and $(x + 2) / (- 1) = (y - 4) /8 = (z - 5) /4$ .
Find the angle between $x /2 = y /2 = z /1$ and $(x - 5) /4 = (y - 2) /1 = (z - 3) /8$ .
Show that $(x - 5) /7 = (y + 2) / (- 5) = z /1$ and $x /1 = y /2 = z /3$ have perpendicular directions.
Find $k$ if directions $(1, k, 2)$ and $(2, - 1, 3)$ are perpendicular.
Find the smaller angle between $(1, 0, 0)$ and $(- 1, 3, 0)$ . Explain why an absolute value is needed.
Two lines have direction vectors $(0, 2, - 3)$ and $(0, - 4, 6)$ . Use proportionality to find their line angle without dividing components.

The numerator dot products are $48 - 36 - 12 = 0$ , $36 + 12 - 48 = 0$ , and $12 - 48 + 36 = 0$ . Dividing by $1 3^{2}$ leaves each at zero. Each triple has squared length one because $144 + 9 + 16 = 169$ , with the same sum in the other triples.

Key Takeaways

• Only direction vectors enter an angle calculation; anchor points do not. • Use the absolute dot product divided by the vector lengths for the smaller line angle. • The cross-product magnitude gives the sine relation and detects parallel directions. • A zero dot product identifies perpendicular directions, while proportional vectors identify parallel directions. • Rewrite nonstandard Cartesian numerators before reading directions; direction tests alone do not prove intersection.

Previous · Lesson 2

Representing a Line in Space

Next · Lesson 4

Shortest Distance Between Lines

Bring the directions together

The dot-product formula

Why the sine formula has squared differences

Parallel and perpendicular directions

Check your understanding

Quiz

Practice Problems

Key Takeaways