Space and time
Vectors: components, addition & products
Drag the coloured dots at the vector tips; or click the diagram and use the arrow keys.
Vector A
A = (3, 4) = 3i + 4j
Results
Magnitude |A|5
Components(3, 4)
Unit vector Â(0.6, 0.8)
Direction: 53.1° north of east (θ = 53.1° from +x) · bearing 36.9°
Working
|A| = √(3² + 4²) = 5
Reference angle α = tan⁻¹(|Ay| / |Ax|) = tan⁻¹(4 / 3) = 53.1°. The vector is in quadrant I, so θ = 53.1° from +x.
Ax = |A| cosθ = 5 × cos 53.1° = 3
Ay = |A| sinθ = 5 × sin 53.1° = 4
📋 Teacher notes
- •A vector has a magnitude (size) and a direction. Its components are its shadows on the axes: Ax = |A| cos θ and Ay = |A| sin θ.
- •From components back to the vector: |A| = √(Ax² + Ay² [+ Az²]). The direction is θ = tan⁻¹(Ay/Ax), but always check the quadrant from the signs of Ax and Ay.
- •The unit vector  = A/|A| has magnitude 1 and carries only the direction, so A = |A| Â. The unit vectors along the axes are i, j and k.
- •A free vector is defined by magnitude and direction only: it is the same vector wherever you draw it.
✏️ Exercises
Curriculum
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