What it is
A vector describes a displacement, or a quantity, that has both magnitude and direction. It is drawn as an arrow. In component form it is written (x, y), where x is the move in the x-direction and y in the y-direction. The vector (3, 4) is 3 to the right and 4 up.
Two vectors are equal if they have the same length and direction, wherever the arrow starts. The arrow from (0, 0) to (3, 4) and the arrow from (1, 1) to (4, 5) are the same vector (3, 4).
Examples:
- Displacement: 3 m east and 4 m north is (3, 4), of length √(3² + 4²) = 5 m.
- Force: 10 N to the right is a vector, because the direction matters.
- Velocity: 50 km/h north is a vector; the speed 50 km/h is only its magnitude.
- Between two points: from A(1, 2) to B(4, 6) the vector is (4 − 1, 6 − 2) = (3, 4).
What is not a vector:
- Temperature, mass and time have size only. 20 °C or 5 kg has no direction and is called a scalar.
- A point (3, 4) gives a place; the vector (3, 4) gives a displacement. They are written the same way but are different things.
- A number such as the length 5 is not the vector (3, 4), even though it is its length. The vectors (3, 4) and (4, 3) have the same length but different directions, so they are different vectors.
Why it matters
Vectors are the language for anything with direction: forces and motion in physics, displacements in geometry. They are calculated with in component form:
- Magnitude: |(3, 4)| = 5.
- Addition: (2, 1) + (1, 3) = (3, 4).
- Scalar multiplication: 2 · (2, 1) = (4, 2).
- Dot product: (2, 1) · (1, 3) = 2 · 1 + 1 · 3 = 5.
- Angle between two vectors, from the dot product.
- Vector projection.
Common misconceptions
- “The vector is the arrow at that exact place.” The arrow can be moved without changing the vector, as long as length and direction are kept.
- “The length of a + b is the length of a plus the length of b.” Not in general. (3, 0) + (0, 4) = (3, 4) has length 5, not 3 + 4 = 7. Equality holds only when the vectors point the same way.
- “A negative vector has negative length.” Length is never negative. −(3, 4) = (−3, −4) points the opposite way and has length 5.
Common questions
What is the difference between a vector and a scalar?
A scalar is an ordinary number, such as 5 kg. A vector also has direction, such as 5 N to the right.
How do I find the vector between two points?
Subtract the start point from the end point, coordinate by coordinate. From (1, 2) to (4, 6) it is (3, 4).
Are there vectors in three dimensions?
Yes, with three components (x, y, z). The length is √(x² + y² + z²).