TL;DR:
A vector has a size (magnitude) and a direction. Add vectors by adding their components.
3 units east plus 4 units north is the vector
(3, 4), with magnitude√(3² + 4²) = 5.The direction is
tan⁻¹(4/3) ≈ 53.1°north of east.
Walk 3 blocks east, then 4 blocks north. How far are you from the start, and in which direction? The answer is not 7 blocks. Distances and directions combine differently from plain numbers, and vectors are the tool that handles both at once.
Size and direction together
A scalar is just a size, such as a speed of 5 m/s. A vector has a size and a direction, such as a velocity of 5 m/s toward the north-east. Forces, velocities and displacements are vectors. We draw them as arrows: the length shows the magnitude and the arrow points in the direction.
In coordinates, a vector is a pair of components: how far it goes along x and along y. The walk above is (3, 4): 3 in the x direction (east) and 4 in the y direction (north).
Worked example: 3 units east plus 4 units north
Write each as components: east 3 is
(3, 0), north 4 is(0, 4).Add component by component:
(3 + 0, 0 + 4) = (3, 4).Magnitude, by the Pythagorean theorem:
√(3² + 4²) = √25 = 5.Direction: the angle above the east axis satisfies
tan θ = 4/3, soθ = tan⁻¹(4/3) ≈ 53.1°.
You end up 5 units from the start, in a direction 53.1° north of east. Check the direction with another ratio: sin θ = 4/5 = 0.8, and sin 53.1° ≈ 0.80. Also cos θ = 3/5 = 0.6, and cos 53.1° ≈ 0.60. All three agree.
Tip to tail
The picture for addition is to draw the first arrow, then start the second where the first ends. The sum runs from the very start to the very end. For our walk, the east arrow and north arrow form two sides of a right triangle, and the sum is the hypotenuse. The order does not matter: north first, then east, lands at the same place.
A non-perpendicular example
Add A = (2, 3) and B = (4, −1). The sum is (2 + 4, 3 + (−1)) = (6, 2). Magnitude: √(36 + 4) = √40 ≈ 6.32. Direction: tan⁻¹(2/6) ≈ 18.4° above the x axis. Notice the magnitudes |A| = √13 ≈ 3.61 and |B| = √17 ≈ 4.12 add to 7.73, which is bigger than 6.32. The sizes do not add unless the vectors point the same way. The triangle inequality says |A + B| ≤ |A| + |B|.
A real use: a boat in a current
A boat heads straight across a river at 4 m/s relative to the water, while the current carries it downstream at 3 m/s (invented values). Relative to the bank, its velocity is the sum: (4, 3) if x is across and y is downstream. The speed over the ground is √(16 + 9) = 5 m/s. It drifts at an angle of tan⁻¹(3/4) ≈ 36.9° from the across direction. If the river is 100 m wide, the crossing takes 100 / 4 = 25 s, and the boat drifts 3 × 25 = 75 m downstream. Check by Pythagoras: the path length is √(100² + 75²) = √(10,000 + 5,625) = √15,625 = 125 m, and 5 × 25 = 125 m. It agrees.
From magnitude and angle to components
Often a vector is given as a size and an angle. A force of 10 N pulls at 30° above the horizontal. The horizontal component is 10 cos 30° ≈ 10 × 0.866 = 8.66 N and the vertical component is 10 sin 30° = 10 × 0.5 = 5 N. Check by rebuilding the magnitude: √(8.66² + 5²) = √(75 + 25) = √100 = 10. Splitting a force into components is the first move in most statics and motion problems, because each direction can then be handled on its own.
Two forces balance when their sum is the zero vector. A pull of (3, 4) N and a pull of (−3, −4) N add to (0, 0), so the object stays still. Check each component: 3 − 3 = 0 and 4 − 4 = 0.
Scaling a vector
Multiplying a vector by a number scales its size. Double (3, 4) to get (6, 8): the magnitude becomes 10, and the direction is unchanged. Multiplying by a negative number reverses the direction: −1 × (3, 4) = (−3, −4). Subtracting a vector is adding its negative.
How to check your answer
Triangle inequality. The sum cannot be longer than the two magnitudes added.
Sketch. Does the arrow for your answer point in a direction that makes sense?
Quadrant. For a direction from inverse tangent, check the signs of the components. A calculator's
tan⁻¹only gives angles from −90° to 90°.Components. Recompute the components from the magnitude and angle:
5 cos 53.1° ≈ 3,5 sin 53.1° ≈ 4.
Common mistakes
Adding magnitudes: 3 + 4 = 7 instead of 5.
Mixing up the order of the components in
tan⁻¹, using x over y.Ignoring the quadrant when x is negative.
Adding a vector to a scalar.
Forgetting units: components must share the same unit.
Where you will use it next
The right-triangle step uses the Pythagorean theorem and the ratios from SOH CAH TOA. Moving and turning many vectors at once is the job of matrices: what multiplying them means. A line is also described by a point and a direction vector, linking back to graphing a line from its equation.
Frequently asked questions
Can two vectors of different sizes add to zero?
Not if they are in the same line with different sizes. They need equal magnitudes and opposite directions, such as (3, 4) and (−3, −4).
Is a vector the same as a point?
A point is a location. A vector is a displacement. The pair (3, 4) can mean either, but a vector can be moved anywhere without changing it.
Get started
