Vectors

Vector subtraction

A - B has magnitude 7 and makes 38.21° with A, for 8 and 5 at 60 degrees.

Vectors

Drag the glowing tips, or type values. Angles are measured from the x axis, and θ is the angle from A to B. Pick a result card to see it built in the picture.

Angle between60°cos θ = A·B / AB

Unit vectors1iB = 0.5i + 0.87j

Vector A of size 8 and vector B of size 5 with 60 degrees between them. The resultant is 11.36, 22.41 degrees from A. The dot product is 20 and the cross product has size 34.64.

Show the formulas

Resultant: R = √(A² + B² + 2AB cos θ) = √(8² + 5² + 2 × 8 × 5 × cos 60°) = 11.36
Direction: tan β = B sin θ / (A + B cos θ) = 5 × sin 60° / (8 + 5 × cos 60°), so β = 22.41° from A
Difference: |A - B| = √(A² + B² - 2AB cos θ) = √(8² + 5² - 2 × 8 × 5 × cos 60°) = 7
Dot product: A·B = AB cos θ = 8 × 5 × cos 60° = 20
Cross product: |A×B| = AB sin θ = 8 × 5 × sin 60° = 34.64
Components: A·B = AxBx + AyBy + AzBz = 8 × 2.5 + 0 × 4.33 + 0 × 0 = 20
Determinant: A×B = (AyBz - AzBy) i + (AzBx - AxBz) j + (AxBy - AyBx) k = 34.64k
Unit vector: Â = A / |A| = 1i

Working

Step by step

Given A = 8, B = 5 and θ = 60°.
|A - B|² = A² + B² - 2AB cos θ = 8² + 5² - 2 × 8 × 5 × cos 60° = 49
|A - B| = √49 = 7
A - B makes 38.21° with A. B - A has the same size and points the opposite way.

Derivation

Why it works

Subtracting a vector means adding its opposite: A - B = A + (-B).
-B has the same length as B and points the opposite way, so the angle between A and -B is 180° - θ.
The resultant formula with θ replaced by 180° - θ gives |A - B|² = A² + B² + 2AB cos(180° - θ).
Since cos(180° - θ) = -cos θ, this becomes |A - B| = √(A² + B² - 2AB cos θ).
B - A has the same size and the opposite direction, so the order of subtraction matters for direction, not for size.

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