Vector definition
A vector is a quantity that has both size and direction. The size of a vector is called its magnitude, and the direction tells us where it points or moves. Vectors are often drawn as arrows. The length of the arrow shows the magnitude, and the arrowhead shows the direction. For example, a movement of 5 metres east is a vector because it has a size and a direction.
A vector is different from a scalar. A scalar has size only, such as mass, time or temperature. A vector has size and direction, such as displacement, velocity or force. For example, saying a car travels at 40 miles per hour gives a speed, which is a scalar. Saying it travels at 40 miles per hour north gives velocity, which is a vector because direction is included.
Vectors are used in geometry, transformations, coordinates, mechanics, physics, engineering, navigation, computer graphics, maps and real-life movement. At GCSE level, students use column vectors to describe translations and movement on grids. At A-Level, vectors are used to describe lines, forces, velocities, accelerations, positions and three-dimensional geometry.
A column vector can describe movement on a coordinate grid. The top number gives the horizontal movement, and the bottom number gives the vertical movement. For example, a vector with top number 3 and bottom number 2 means move 3 to the right and 2 up. If the top number is negative, move left. If the bottom number is negative, move down.
Vectors can be added. If one vector moves 2 right and 1 up, and another vector moves 3 right and 4 down, the combined movement is 5 right and 3 down. This works because the horizontal parts are added together and the vertical parts are added together. Vector addition is useful for combining movements, forces or journeys.
A scalar multiple of a vector changes its magnitude. For example, multiplying a vector by 2 doubles its length while keeping the same direction. Multiplying by -1 reverses the direction. This idea is important when comparing parallel vectors, describing repeated movement or solving geometry problems involving ratios and line segments.
Vectors are closely linked to translation. A translation moves every point of a shape by the same vector. This means the vector describes the exact movement from the object to the image. In mechanics, vectors describe quantities such as force and velocity because direction matters. A force of 10 newtons to the right is different from a force of 10 newtons to the left.
Related ideas include magnitude, direction, scalar, column vector, displacement, velocity, force, coordinates, translation and geometry. A clear definition of vector helps students understand that direction is just as important as size. In exams, read the direction carefully, keep horizontal and vertical components in the correct order, and remember that opposite directions may require negative values.