Kuzee
Vector Algebra and Analytic Geometry of Space

Angle Formulas

The Angle between Two Lines

The cosine formula for the angle between two lines

Let the direction vectors of the two lines be s⃗1\vec{s}_1 and s⃗2\vec{s}_2. The cosine of the angle is:

cos⁡θ=∣s⃗1⋅s⃗2∣∣s⃗1∣∣s⃗2∣\cos\theta = \frac{|\vec{s}_1 \cdot \vec{s}_2|}{|\vec{s}_1| |\vec{s}_2|}

The Range of the Angle

The angle θ\theta between two lines ranges over [0∘,90∘][0^\circ, 90^\circ], that is, the acute angle is taken.

Special Cases

  • when cos⁡θ=0\cos\theta = 0, the two lines are perpendicular
  • when cos⁡θ=1\cos\theta = 1, the two lines are parallel in the same direction
  • when cos⁡θ=−1\cos\theta = -1, the two lines are parallel in opposite directions

The Angle between a Line and a Plane

The sine formula for the angle between a line and a plane

Let the direction vector of the line be s⃗\vec{s} and the normal vector of the plane be n⃗\vec{n}. The sine of the angle is:

sin⁡θ=∣s⃗⋅n⃗∣∣s⃗∣∣n⃗∣\sin\theta = \frac{|\vec{s} \cdot \vec{n}|}{|\vec{s}| |\vec{n}|}

The Definition of the Angle

The angle between a line and a plane

The angle between a line and a plane is the angle between the direction vector s⃗\vec{s} of the line and the plane (taking the acute angle); it is complementary to the angle between the direction vector s⃗\vec{s} and the normal vector n⃗\vec{n}, and so it is computed with the sine formula sin⁡θ=∣s⃗⋅n⃗∣∣s⃗∣ ∣n⃗∣\sin\theta = \dfrac{|\vec{s}\cdot\vec{n}|}{|\vec{s}|\,|\vec{n}|}.

The Geometric Meaning

  • θ=0∘\theta = 0^\circ: the line is parallel to the plane
  • θ=90∘\theta = 90^\circ: the line is perpendicular to the plane

The Angle between Two Planes

The cosine formula for the angle between two planes

Let the normal vectors of the two planes be n⃗1\vec{n}_1 and n⃗2\vec{n}_2. The cosine of the angle is:

cos⁡θ=∣n⃗1⋅n⃗2∣∣n⃗1∣∣n⃗2∣\cos\theta = \frac{|\vec{n}_1 \cdot \vec{n}_2|}{|\vec{n}_1| |\vec{n}_2|}

The Range of the Angle

The angle θ\theta between two planes ranges over [0∘,90∘][0^\circ, 90^\circ].

Special Cases

  • when cos⁡θ=0\cos\theta = 0, the two planes are perpendicular
  • when cos⁡θ=1\cos\theta = 1, the two planes are parallel

Methods of Computation

The Coordinate Method

Compute the scalar product from the coordinates of the direction vectors or the normal vectors:

a⃗⋅b⃗=axbx+ayby+azbz\vec{a} \cdot \vec{b} = a_x b_x + a_y b_y + a_z b_z

The Geometric Method

Use geometric relationships and the properties of trigonometric functions.

Techniques for Applying Them

  1. Unify to unit vectors: normalise the vectors before computing
  2. Choose a coordinate system wisely: pick one that makes the computation easier
  3. Use known relationships: use known perpendicularity and parallelism to simplify the computation

Exercises

Exercise 1

Find the angle between the two planes x+y+z=1x + y + z = 1 and x−y+z=2x - y + z = 2.

the angle between two planesthe normal vectorsthe cosine formula

Approach:

Compute the angle between the normal vectors of the two planes.

Steps:

  1. The normal vectors: (1,1,1)(1, 1, 1) and (1,−1,1)(1, -1, 1)
  2. cos⁡θ=∣1×1+1×(−1)+1×1∣3×3=∣1−1+1∣3=13\cos\theta = \frac{|1 \times 1 + 1 \times (-1) + 1 \times 1|}{\sqrt{3} \times \sqrt{3}} = \frac{|1 - 1 + 1|}{3} = \frac{1}{3}
  3. θ=arccos⁡13\theta = \arccos\frac{1}{3}

Answer: the angle is arccos⁡13\arccos\frac{1}{3}

Exercise 2

Find the angle between the line x1=y1=z1\frac{x}{1} = \frac{y}{1} = \frac{z}{1} and the plane x+y+z=1x + y + z = 1.

the angle between a line and a planethe direction vectorthe normal vector

Approach:

Compute the angle between the direction vector of the line and the normal vector of the plane.

Steps:

  1. The direction vector of the line: (1,1,1)(1, 1, 1)
  2. The normal vector of the plane: (1,1,1)(1, 1, 1)
  3. sin⁡θ=∣(1,1,1)⋅(1,1,1)∣3×3=33×3=33=1\sin\theta = \frac{|(1,1,1) \cdot (1,1,1)|}{\sqrt{3} \times \sqrt{3}} = \frac{3}{\sqrt{3} \times \sqrt{3}} = \frac{3}{3} = 1
  4. θ=90∘\theta = 90^\circ

Answer: the angle is 90∘90^\circ (the line is perpendicular to the plane)

Summary

Symbols Used in This Article

SymbolTypePronunciation/ExplanationMeaning in This Article
θ\thetaAnglethetaThe angle between geometric objects
cos⁡θ\cos\thetaCosinecosine thetaThe cosine of the angle
sin⁡θ\sin\thetaSinesine thetaThe sine of the angle
s⃗\vec{s}Direction vectordirection vectorThe direction vector of the line
n⃗\vec{n}Normal vectornormal vectorThe normal vector of the plane
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