A negative dot product gives an obtuse angle between the two chosen direction vectors. If the question asks for the smaller angle between the lines, take the supplement of that obtuse angle. Reversing one direction vector leaves its line unchanged.
SEAB’s 2027 A-Level H2 Mathematics syllabus (9758) includes scalar products and the angle between two lines. The worked examples below show how the wording decides whether the sign of the dot product stays in the cosine calculation.
- Read whether the question asks about vectors or lines.
- Use the direction vectors from the line equations.
- Use the absolute dot product for the smaller line angle.
Worked example | 120° between vectors, 60° between lines
Let the direction vectors of two lines be u = (1, 1, 0) and v = (−1, 0, 1). Their dot product is −1, and each has magnitude √2. The cosine of the angle between u and v is −1 ÷ (√2 × √2) = −1/2, so that angle is 120°.
For the smaller angle between the lines, use 180° − 120° = 60°. You can also use |u · v| ÷ (|u||v|) = 1/2 from the start when the question asks for the smaller line angle.
Replace v by −v = (1, 0, −1). It points the other way along the same line. Its dot product with u is +1, so the angle between this pair of vectors is already 60°. The smaller line angle agrees.
Which values belong in the formula?
If the lines are written as r = a + λu and r = b + μv, use u and v. The vectors a and b locate points on the lines. Their components do not specify the lines’ directions.
If the question names directed vectors such as AB and AC, use their signed dot product to find the angle between those vectors. Taking the absolute value automatically would replace an obtuse angle with its acute supplement.
Check an example with unequal magnitudes
Take direction vectors (1, 0, 0) and (−3, 4, 0). Their dot product is −3 and the product of their magnitudes is 5. The directed-vector angle is cos⁻¹(−3/5) ≈ 126.87°. The smaller line angle is cos⁻¹(3/5) ≈ 53.13°.
Ask a study partner to explain why reversing either direction vector can change the first result and leaves the second unchanged. Then test perpendicular directions, where the dot product is zero and the angle is 90° either way.
Frequently asked questions
Can the smaller angle between two lines be 90°?
Yes. Perpendicular directions give a zero dot product.
Do skew lines need to intersect for this calculation?
Their directions still determine an angle. Use the direction vectors without moving the lines themselves.
Official information
SEAB’s 2027 A-Level H2 Mathematics syllabus (9758), Sections 3.2 Scalar and Vector Products and 3.3 Three-Dimensional Vector Geometry. Checked 9 November 2026.