Vectors in 3 Dimensions (CIE A Level Maths: Pure 3)

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Vectors in 3 Dimensions

What is a 3-D vector?

  • Vectors represent a movement of a certain magnitude (size) in a given direction

    You 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

Vectors in 3 Dimensions Diagram 1, AS & A Level Maths revision notes

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)

 

Vectors in 3 Dimensions Diagram 2a, AS & A Level Maths revision notes

  • 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:

Vectors in 3 Dimensions Diagram 2b, AS & A Level Maths revision notes

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

 Vectors in 3 Dimensions Diagram 3, AS & A Level Maths revision notes

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

Vectors in 3 Dimensions Diagram 4a, AS & A Level Maths revision notes

 

Vectors in 3 Dimensions Diagram 4b, AS & A Level Maths revision notes

Worked example

Vectors in 3 Dimensions Example Solution, AS & A Level Maths revision notes

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Lucy

Author: Lucy

Expertise: Head of STEM

Lucy has been a passionate Maths teacher for over 12 years, teaching maths across the UK and abroad helping to engage, interest and develop confidence in the subject at all levels. Working as a Head of Department and then Director of Maths, Lucy has advised schools and academy trusts in both Scotland and the East Midlands, where her role was to support and coach teachers to improve Maths teaching for all. Lucy has created revision content for a variety of domestic and international Exam Boards including Edexcel, AQA, OCR, CIE and IB.