AQA A Level Maths: Pure复习笔记11.2.1 Vectors in 3 Dimensions

Vectors in 3 Dimensions

What is a 3-D vector?

  • Vectors represent a movement of a certain magnitude (size) in a given directionYou should have already come across (2D) vectors at AS (see Basic Vectors)
  • 3-D vectors describe the position of a point in a 3-D space in relation to the origin
  • They can be represented in different ways such as a column vector or in i, j, k unit vector form


Magnitude of a 3-D vector

  • The magnitude of a 3-D vector is simply its size
  • Like 2-D vectors we can find the magnitude using Pythagoras’ theorem (see Magnitude Direction)


  • For 3-D position vectors we can find the distance between two points
  • By using the respective co-ordinates we can calculate the magnitude of the vector between them:


3-D vector addition, scalars, parallel vectors and unit vectors

  • 3-D vectors work in the same way as 2-D vectors, just in three dimensions rather than two
  • Vector addition and subtraction and scalar multiplication can be carried out in exactly the same way, this time involving i, j and k or x, y and z
  • 3-D vectors are also parallel if one is a multiple of the other


  • Unit vectors in 3-D are found in exactly the same way as in 2-D



Worked Example