medium difficulty8 min

Angle Between Two Lines in Space

The angle between two lines in space is the angle between their direction vectors. If the direction vectors are a₁ and a₂, the acute angle θ between the lines is found using their dot product.

In one line: Dot product measures how much two directions face each other.

Practice This Concept

Understand it

The position of the lines does not affect the angle; only their directions matter.
For direction vectors a₁ and a₂, cos θ = |a₁ · a₂|/(|a₁||a₂|) gives the acute angle.
If the dot product is 0, the lines are perpendicular.
If the direction ratios are proportional, the lines are parallel.

Example

For lines with direction vectors 2i + j - 2k and i + 2j + 2k, the dot product is 2(1) + 1(2) + (-2)(2) = 0, so the lines are perpendicular.

Formula and rule

Angle formula: cos θ = |a₁ · a₂|/(|a₁||a₂|).

Perpendicular condition: a₁ · a₂ = 0.

Parallel condition: a₁, a₂, a₃ proportional to b₁, b₂, b₃, meaning a₁/b₁ = a₂/b₂ = a₃/b₃ where ratios are defined.

See it worked out

Follow the example line by line, then use the same method on new questions.

Worked example

Find the acute angle between lines with direction ratios (1, -2, 2) and (3, 0, 4).

1Dot product = 1(3) + (-2)(0) + 2(4) = 11.
2Magnitudes: √(1² + (-2)² + 2²) = 3 and √(3² + 0² + 4²) = 5.
3Therefore cos θ = |11|/(3 × 5) = 11/15.
4Hence θ = cos⁻¹(11/15).

Method you can reuse

1Extract direction ratios from each line.
2Form the dot product.
3Find both magnitudes.
4Substitute in cos θ = |dot product| divided by product of magnitudes.
5For perpendicularity, stop once the dot product is 0.
6For parallelism, check proportional direction ratios.

Visual tip

A diagram is not essential for calculation because the angle is found from direction vectors, but a small sketch can show that skew lines may still have an angle through their directions.

Check yourself

Answer the question in your head before opening the answer.

Try it yourself

Two lines have direction ratios (1, 2, 2) and (2, -1, 0). Are they perpendicular?

Think first, then show answer

Dot product = 1(2) + 2(-1) + 2(0) = 0, so the lines are perpendicular.

Common wrong answer

Dropping the modulus in the acute angle formula and reporting an obtuse angle when the question asks for the angle between two lines.

Don't mix this up

Students sometimes use points on the lines instead of direction vectors in the dot product formula.

Exam tip

For angle questions, ignore the fixed point of each line after confirming the direction vectors. The direction vectors alone decide the angle.

Write it in the exam

Each point is worth about one mark. Write point 1 for a 1-mark question and keep adding points as the marks go up.

How CBSE asks it

Asked as direct angle calculation, checking perpendicularity or parallelism, or as part of a line-and-plane or shortest-distance problem where direction vectors must first be extracted.

1
Enough for 1 mark
The angle between two lines in space is the angle between their direction vectors. If the direction vectors are a₁ and a₂, the acute angle θ between the lines is found using their dot product.
2
Add for 2 marks
Example: For lines with direction vectors 2i + j - 2k and i + 2j + 2k, the dot product is 2(1) + 1(2) + (-2)(2) = 0, so the lines are perpendicular.
3
Add for 3 marks
The position of the lines does not affect the angle; only their directions matter.

MCQ Quiz

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