What is a vector?
A scalar is a quantity with size only, such as mass (5 kg) or temperature (20 °C). Avector has both magnitude (size) and direction, such as a displacement of 5 km north-east or a force of 10 N downward. We draw a vector as an arrow: its length shows the magnitude and its orientation shows the direction. Moving an arrow without rotating or resizing it does not change the vector.
Adding vectors graphically
- Tip-to-tail method: draw A, then start B at the tip of A. The resultant runs from the tail of A to the tip of B.
- Parallelogram method: draw A and B from the same starting point and complete the parallelogram. The diagonal from the shared start is the resultant.
Both methods always give the same answer, and the order does not matter: A + B = B + A. The size of the resultant depends on the angle between the vectors — largest when they point the same way, smallest when they point in opposite directions.
Adding with components
Any vector can be split into an x-component and a y-component — how far it goes across and how far up. Adding vectors then becomes simple arithmetic: add all the x-components, add all the y-components, and rebuild the resultant from the totals. This is how scientists and engineers add many forces or velocities at once.
Subtracting vectors
The negative of a vector, −B, has the same length as B but points the opposite way. To subtract, add the negative: A − B = A + (−B). When A and B start at the same point, A − B is the arrow from the tip of B to the tip of A. This is how you find a change in velocity or the position of one object relative to another.
Formulas
Components: vₓ = |v| cos θ v_y = |v| sin θSum: R = (Aₓ + Bₓ, A_y + B_y)Magnitude: |R| = √(Rₓ² + R_y²) Direction: θ = tan⁻¹(R_y / Rₓ) (check the quadrant)For two vectors with angle φ between them, the law of cosines gives:
|A + B|² = |A|² + |B|² + 2|A||B| cos φWorked example
A student walks 3 km east and then 4 km north. What is their displacement from the start?
- Write each step as components: a = (3, 0) and b = (0, 4), with x east and y north.
- Add the components: a + b = (3 + 0, 0 + 4) = (3, 4).
- Find the magnitude with Pythagoras: |a + b| = √(3² + 4²) = √25 = 5 km.
- Find the direction: θ = tan⁻¹(4 / 3) ≈ 53.1° north of east.
The student walked 7 km in total, but their displacement is only 5 km. Distance adds up every step; displacement is a vector, so steps in different directions partly cancel. Try it in the simulator by setting the two vectors to (3, 0) and (0, 4).
Key terms
- Magnitude
- The size or length of a vector, always positive or zero.
- Resultant
- The single vector that has the same effect as two or more vectors combined.
- Component
- The part of a vector along one axis.
- Unit vector
- A vector of length 1 that indicates a direction only.
Real-world applications
- Navigation: pilots and sailors add their own velocity to the wind or current velocity to find their true path over the ground.
- Forces: engineers add all the forces on a bridge or building to check that they balance.
- Sport: a swimmer crossing a river ends up downstream because their velocity adds to the river’s.
- Computer graphics and games: positions, movement and lighting are calculated with vector arithmetic.
Common misconceptions
- “The resultant’s length is the sum of the lengths.” Only when the vectors point in exactly the same direction.
- “Moving an arrow changes the vector.” A vector is defined by size and direction, not position.
- “tan⁻¹ always gives the right angle.” It only returns angles between −90° and 90°; you must check which quadrant the vector is in.
Key takeaways
- Vectors have magnitude and direction and add tip-to-tail.
- Adding components separately is the most reliable method.
- The resultant is between |A| − |B| and |A| + |B|, depending on the angle.
- Subtraction means adding the opposite vector.