CIE A Level Maths: Pure 3复习笔记7.2.1 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

11.2.1-Vectors-in-3-Dimensions-Diagram-1

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)

11.2.1-Vectors-in-3-Dimensions-Diagram-2a

  • 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:

11.2.1-Vectors-in-3-Dimensions-Diagram-2b

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

 11.2.1-Vectors-in-3-Dimensions-Diagram-3

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

11.2.1-Vectors-in-3-Dimensions-Diagram-4a

11.2.1-Vectors-in-3-Dimensions-Diagram-4b

Worked Example

11.2.1-Vectors-in-3-Dimensions-Example-Solution

 

 

 

 

 

 

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