Vectors (Higher Tier)

Basic Concepts

A vector represents a quantity that has both magnitude (size) and direction, like velocity or force.
Vectors are typically represented with an arrow. The length denotes its magnitude and the arrowhead denotes its direction.
A position vector represents the relative position of a point from a reference point (usually denoted by the origin).
In two dimensions, a vector can be represented as (x, y), which can also be written as xi + yj.
The magnitude of a vector (x, y) is calculated by the formula √(x² + y²).

Operations with Vectors

Vector addition is performed component-wise, i.e., the x components are added together, and the y components are added together. For example, two vectors (x1, y1) and (x2, y2) are added to give (x1 + x2 , y1 + y2).
Vector subtraction is also performed component-wise. A vector (x1, y1) subtracted from another vector (x2, y2) would result in (x2 - x1, y2 - y1).
A vector (x, y) can be scaled by multiplying both its components with a scalar value. A vector scaled by a scalar k would be (kx, ky).

Dot Product

The dot product of two vectors (x1 , y1) and (x2 , y2) is calculated as (x1 * x2) + (y1 * y2).
The dot product is a scalar quantity, not a vector.
It is used to find the cosine of the angle between two vectors, and to identify perpendicular vectors (as the dot product of perpendicular vectors is zero).

Vector Equations

A vector equation represents a line or a plane in vector form.
For instance, a line passing through a point A with position vector a and parallel to a vector b can be represented as r = a + tb.
t is a scalar, known as the parameter of the line. Different values of t give different points on the line.

Cross Product

The cross product of two vectors results in a new vector that is perpendicular to both original vectors.
The magnitude of the cross product vector is equal to the area of the parallelogram formed by the two original vectors.

These points encapsulate the main ideas of vectors at the Higher Tier level. Practise regularly to become familiar with these concepts. Challenge your understanding by solving different problems involving vectors.