Maths
Vectors
Learn A-Level vectors in two and three dimensions, including column vectors, magnitude and direction, vector addition, position vectors, distance, resultant force and kinematics.
Vectors are quantities with both magnitude and direction and are used in A-Level maths to represent displacement, velocity, force and other directed quantities.
Vectors can be represented in two dimensions or three dimensions using component or column vector form.
Column vectors show the horizontal, vertical and, in three dimensions, depth components of a vector in a compact mathematical form.
The magnitude of a vector is its length and in two dimensions can be found using Pythagoras from its horizontal and vertical components.
The direction of a vector can be found from its components using trigonometric relationships, allowing conversion between component form and magnitude-direction form.
A unit vector has magnitude 1 and is useful for describing direction independently of the size of a vector.
Vector addition is performed by adding corresponding components, and it can also be represented geometrically using a vector diagram.
Vector subtraction is carried out by subtracting corresponding components and can be interpreted as adding the negative of a vector.
Multiplying a vector by a scalar changes its magnitude and may reverse its direction if the scalar is negative.
Scalar and vector quantities differ because a scalar has magnitude only, while a vector quantity has both magnitude and direction.
A position vector gives the position of a point relative to the origin and can be used to describe points in two-dimensional and three-dimensional space.
The vector between two points is found by subtracting their position vectors, and its magnitude gives the distance between the two points.
Vectors can be used to combine forces, and the resultant force is the single vector that has the same overall effect as all the individual force vectors acting together.
Force and resultant force problems can be solved by resolving vectors into components and then adding the components in each direction.
Vectors are widely used with equations of motion in kinematics to represent displacement, velocity and acceleration and to solve mathematical problems in context.