Vectors in the Plane

A vector is a line segment running from point A (tail) to point B (head).

Vector

Each vector has a magnitude (also referred to as length) and a direction.

Direction of a Vector

This is the direction of the line which contains the vector or any line which is parallel to it.

Magnitude of a Vector





Vector

 

The magnitude of the vector Vector is the length of the line segment Vector. It is denoted by Vector Magnitude.

The magnitude of a vector is always a positive number or zero.

Magnitude of a Vector

Magnitude of a Vector Formula

The magnitude of a vector can be calculated if the coordinates of the endpoints are known:

Magnitude of a Vector

Magnitude of a Vector Formula

Examples

Calculate the magnitude of the following vectors:

Vector Calculations

Vector Solution


Calculate the value of k knowing the magnitude of the vector vector= (k, 3) is 5.

Vector Solution

Position Vector

Position Vector

The vector Vector that joins the coordinates origin, O, with a point, P, is the position vector of the point P.

Components or Coordinates of a Vector

Componets and Coordinates of a Vector

If the coordinates of A and B are:

CoordinatesCoordinates

Magnitude of a Vector Formula

Examples

Find the components of the vector :

Vector Calculations


The vector vector has the components (5, −2). Find the coordinates of A if the terminal point is known as B(12, −3).

Vector Calculations


Calculate the coordinates of Point D so that the quadrilateral of Points: A(−1, −2), B(4, −1), C(5, 2) and D form a parallelogram.

ParallelogramVector Calculations