Vectors
A short lesson with diagrams, a worked example, a problem to try, and a quick check. Then quiz yourself on 5 practice questions.
A vector has a size and a direction. To add vectors, break each one into x and y components, add the x’s together and the y’s together, then put the result back together with a right triangle.
1. Arrows and tip-to-tail
Draw a vector as an arrow: its length shows the size (magnitude), and the way it points shows the direction.
To add vectors on paper, put the tail of each new arrow on the tip of the last one. The resultant runs from the very first tail to the very last tip.
Here A and B are perpendicular, so R = √(4² + 3²) = 5 m, not 4 + 3 = 7 m.
2. Breaking a vector into components
Components pointing west (−x) or south (−y) are negative. Keep those signs when you add.
3. Example, then try it
A = 10 m at 30° above east. Find its components.
Ax = 10 cos 30° ≈ 8.7 m (east)
Ay = 10 sin 30° = 5.0 m (north)
Check: √(8.7² + 5.0²) ≈ 10 m. ✓
A = 10 m at 37° above east. B = 9.0 m north. Find the size and direction of A + B. (Use cos 37° ≈ 0.80 and sin 37° ≈ 0.60.)
Components of A: Ax = 10(0.80) = 8.0 m; Ay = 10(0.60) = 6.0 m.
Components of B: Bx = 0; By = 9.0 m.
Add each direction: Rx = 8.0 + 0 = 8.0 m; Ry = 6.0 + 9.0 = 15.0 m.
Recombine: R = √(8.0² + 15.0²) = √289 = 17 m; tan θ = 15 ÷ 8 → θ ≈ 62° above east.
Answer: 17 m at about 62° above east. (Not 10 + 9 = 19 m: the vectors point in different directions.)
4. Common mistakes
Vector A is 20 m long at 30° above the +x axis. What is its x-component?
The angle is measured from the x-axis, so Ax = A cos θ = 20 cos 30° ≈ 17 m. The 10 m answer is A sin 30°, the y-component.
Ready to practice?
5 questions on Vectors, each with a hint and an explanation for every wrong answer.