01 · Positive x-axis
Find the angle between (1, 2, 2) and the positive x-axis.
Hint
The magnitude is 3.
Worked solution
cos⁻¹(1/3) ≈ 70.529°.
Understand · explore · practise
Angles with axes and planes
Find the angles a 3D vector makes with positive coordinate axes and distinguish them from its inclination to a coordinate plane. Handle negative components, vertical directions and the zero vector.
Before you startVector magnitudes, right-triangle trigonometry and inverse cosine in degree mode.
01 / Specify the axis or plane before choosing a formula
An axis has a positive direction; a plane has two sides.
cos α = x/|v|, cos β = y/|v|, cos γ = z/|v|
For a nonzero v = (x, y, z), α, β and γ are angles with the positive x-, y- and z-axes.
Each axis angle lies between 0° and 180°. A negative component gives an obtuse angle to that positive axis. The inclination of a line or vector to a plane is instead taken between 0° and 90°.
An axis angle and a plane angle differExplore
|v| = √41 ≈ 6.403124; horizontal projection length = 5.
Angles with +x, +y, +z: 62.061647°, 51.340192°, 51.340192°.
Inclination to the xy-plane = 38.659808°.
Blue: v in its vertical cross-section. Amber: horizontal projection length. Green: signed height. The drawing collapses the horizontal direction; read the three axis angles from the components. A plane inclination is between 0° and 90°, regardless of whether v points above or below it.
02 / Use a component divided by the full length
A right-triangle cross-section gives the ratio.
Find the angle of v = (2, 3, 6) with the positive z-axis.Worked example
|v| = √(4 + 9 + 36) = 7
Use all three components for the hypotenuse.
cos γ = 6/7
Use the component along the specified axis.
γ = cos⁻¹(6/7) ≈ 31.003°
Use degree mode.
03 / Keep the sign of the axis component
An absolute value would lose the orientation.
Find the angle of v = (−2, 1, 2) with the positive x-axis.Worked example
|v| = 3
Magnitude remains positive.
cos α = −2/3
Keep the negative numerator.
α ≈ 131.810°
The angle is obtuse, as the negative x component predicts.
02 · Negative z component
Find the angle of (0, 3, −4) with the positive z-axis.
Hint
Use cos γ = −4/5.
Worked solution
γ ≈ 143.130°. The acute supplementary angle would measure the angle to the negative z-axis instead.
03 · Reverse an axis
If a vector makes angle α with +x, what angle does it make with −x?
Hint
The two directed axes are opposite.
Worked solution
180° − α.
04 / A zero component gives a right angle to that axis
The vector must still be nonzero.
v = (0, 3, 4).Worked example
Angle with +x = 90°
The x component is zero.
Angle with +y = cos⁻¹(3/5)
This is approximately 53.130°.
Angle with +z = cos⁻¹(4/5)
This is approximately 36.870°.
04 · Along a negative axis
State the three positive-axis angles of (0, 0, −5).
Hint
It is perpendicular to x and y, but opposite to +z.
Worked solution
α = 90°, β = 90°, γ = 180°.
05 · Zero vector
Why do these formulas not give angles for (0, 0, 0)?
Hint
The denominator |v| is zero.
Worked solution
The zero vector has no direction, and the ratios are undefined.
05 / The three squared direction cosines sum to one
The axis angles themselves do not normally sum to 90° or 180°.
cos²α + cos²β + cos²γ = 1
Divide x² + y² + z² = |v|² by |v|².
Could a vector make 60° with all three positive axes?Worked example
Each squared cosine would be 1/4
cos 60° = 1/2.
The sum would be 3/4
This is not 1.
No such nonzero vector exists
All three direction ratios must satisfy the identity.
06 · Equal components
Find the three positive-axis angles of (1, 1, 1).
Hint
Each cosine is 1/√3.
Worked solution
All three are approximately 54.736°. Their sum is neither 90° nor 180°.
06 / Project onto the plane, then form a right triangle
The vertical component is opposite the plane inclination.
sin θ = |z|/|v|; tan θ = |z|/√(x² + y²)
For inclination θ to the xy-plane. Use the tangent form only when the horizontal projection is nonzero.
Find the inclination of (3, 4, 4) to the xy-plane.Worked example
Horizontal projection length = √(3² + 4²) = 5
Ignore z only when finding the projection.
tan θ = 4/5
Compare vertical height with horizontal length.
θ ≈ 38.660°
This differs from the angle to the positive z-axis.
Watch: axis angle versus plane inclination
Pause, replay or seek freely. The notes explain the same idea and stay in view.
07 · Below the plane
Find the inclination of (3, 4, −4) to the xy-plane.
Hint
Inclination uses |z|.
Worked solution
tan⁻¹(4/5) ≈ 38.660°, the same inclination as its upward counterpart.
07 / Relate the two angles carefully
θ = |90° − γ| for the xy-plane and positive z-axis.
v = (0, 3, −4).Worked example
Angle to +z is γ ≈ 143.130°
Keep the negative component for the axis angle.
Plane inclination θ = |90° − γ| ≈ 53.130°
Inclination is nonnegative and at most 90°.
sin θ = 4/5
This provides a direct check.
08 · A common confusion
A vector makes 120° with +z. What is its inclination to the xy-plane?
Hint
Use the absolute difference from 90°.
Worked solution
30°, not −30° or 60°.
09 · Parallel to the plane
What is the inclination of (−3, 4, 0) to the xy-plane?
Hint
Its vertical component vanishes.
Worked solution
0°. Its angle to +z is 90°.
08 / A vertical direction is perpendicular to the xy-plane
Avoid dividing by a zero horizontal projection.
v = (0, 0, −5).Worked example
The horizontal projection has length zero
The tangent quotient would divide by zero.
sin θ = 5/5 = 1
The sine form still works.
θ = 90°
Both upward and downward vertical lines are perpendicular to the plane.
10 · Another coordinate plane
Find the inclination of (4, 0, 3) to the yz-plane.
Hint
The perpendicular component to that plane is x.
Worked solution
sin θ = |4|/5, so θ ≈ 53.130°.
11 · The xz-plane
State the sine formula for inclination to the xz-plane.
Hint
The missing coordinate names the perpendicular direction.
Worked solution
sin θ = |y|/|v|, for v ≠ 0.
09 / Recover components using length and direction
Retain signs required by the specified angles.
A vector has magnitude 10 and makes angle 60° with +x. Find its x component.Worked example
x = |v| cos α
Rearrange the direction-cosine formula.
x = 10 cos 60° = 5
A component is not the full vector.
y² + z² = 75
Many vectors satisfy this information.
12 · Two direction cosines
A unit vector has x = 1/3 and y = 2/3. Find its possible z components.
Hint
Use x² + y² + z² = 1.
Worked solution
z = ±2/3. More information is needed to select a sign.
13 · Select the sign
For that unit vector, its angle with +z is obtuse. Which z is correct?
Hint
An obtuse angle has a negative cosine.
Worked solution
z = −2/3.
10 / Find the displacement before finding its angle
The requested direction determines the endpoint order.
A = (1, 2, 0), B = (4, 6, −4). Find the inclination of AB to the xy-plane.Worked example
AB = (3, 4, −4)
Use endpoint minus startpoint.
Horizontal projection length = 5
Use the first two displacement components.
θ = tan⁻¹(4/5) ≈ 38.660°
The downward component changes orientation but not inclination.
14 · Reverse direction
Compare the positive-axis angles and plane inclination of AB and BA.
Hint
Reversing negates all components.
Worked solution
Each positive-axis angle becomes its supplement. Inclination to the plane stays the same.
15 · Check plausibility
Can a nonzero vector have obtuse angles to all three positive axes?
Hint
Consider all three components negative.
Worked solution
Yes. For (−1, −1, −1), each angle is cos⁻¹(−1/√3) ≈ 125.264°.
11 / Choose the correct signed or unsigned ratio
State what the angle is measured against.
- For a positive-axis angle, retain the signed component.
- Use the full three-dimensional magnitude.
- For a plane inclination, use the absolute perpendicular component.
- Check the horizontal projection before using tangent.
- Use degree mode when answers are in degrees.
- The zero vector has no direction angles.
- Use the direction-cosine identity as a consistency check.
16 · Final comparison
For (0, −3, 4), find its angle with +y and its inclination to the xz-plane.
Hint
The same component enters with different sign conventions.
Worked solution
Angle to +y = cos⁻¹(−3/5) ≈ 126.870°. Inclination to xz = sin⁻¹(3/5) ≈ 36.870°.
Section 1 of 11 · Specify the axis or plane before choosing a formula