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Vectors

A vector is a quantity with both magnitude and direction, and is typically represented visually by a line segment between two points.

Vectors, representing vectors, unit vectors, column vectors, parallel vectors, vector triangle, triangle law for vector addition. A-Level Maths Notes, GCSE Maths. EngineeringNotes.net, EngineeringNotes, Engineering Notes

There are many ways of representing vectors in notation:

  • The two points that the vector connects with an arrow above them

  • As a bold typeface lower case letter

  • As an underlined lower case letter

  • As a column vector, showing displacement in the x-direction above that in the y-direction

  • As a multiple of unit vectors, i (one unit in the positive x-direction) and j (one unit in the positive y-direction)

These notes will predominantly use bold typeface of lower case letters, underlined letters and the unit vectors i and j.

Multiplying vectors by scalars, vector multiplication. A-level Maths Notes, GCSE Maths. EngineeringNotes.net, EngineeringNotes, Engineering Notes

Vectors can be multiplied by a scalar, and added and subtracted:

Column vectors, adding and subtracting column vectors, multiplying column vectors by a scalar. A-Level Maths Notes. EngineeringNotes.net, EngineeringNotes, Engineering Notes



Magnitude & Direction

The magnitude of a vector is given by Pythagoras' Theorem. Magnitude is noted using straight lines on either side of the letter, like modulus.

For the vector a = xi + yj, |a| = √(x²+y²)

A unit direction vector, â, can be found as a / |a|


A vector can also be defined by giving its magnitude and the angle it makes with one of the coordinate axis. This is called magnitude-direction form.



Position Vectors

Position vectors are used to give the location of a point relative to a fixed origin.

Position vector, A-Level Maths notes. EngineeringNotes.net, EngineeringNotes, Engineering Notes
A point (p, q) has a position vector pi + qj

Vector Geometry

Position vectors can be used to solve geometric problems:

Geometric problems and vectors, vector problems, A-level maths notes. EngineeringNotes.net, EngineeringNotes, Engineering Notes



Vectors in 3D

In three-dimensional geometry, there are x, y, and z-axes. This means each coordinate has three values, (x, y, z). The position vector of this is xi + yj + zk

3D Vectors, Vectors in 3D, Vectors in three dimensions, A-Level Maths Notes. EngineeringNotes.net, EngineeringNotes, Engineering Notes

Pythagoras' Theorem still applies, but adjusted for three points:

Distance from point (x, y, z) to origin is given as √(x²+y²+z²)
3D Vectors, Vectors in 3D, Vectors in three dimensions, A-Level Maths Notes. EngineeringNotes.net, EngineeringNotes, Engineering Notes

The distance between two points, (x₁, y₁, z₁) and (x₂, y₂, z₂) is given as:

√( (x₁ - x₂)² + (y₁ - y₂)² + (z₁ - z₂)² )

Vector Geometry in 3D

3D vector geometry, three dimensional geometry. A-Level Maths Notes. EngineeringNotes.net, EngineeringNotes, Engineering Notes