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Components of Vectors (DP IB Applications & Interpretation (AI)): Revision Note

Amber

Written by: Amber

Reviewed by: Dan Finlay

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Components of vectors

What are components of a vector?

  • Components of a vector describe how the vector acts in different directions

    • Consider 4 bold italic i minus 3 bold italic j plus bold italic k as an example

      • The component in the bold italic i direction is 4

      • The component in the bold italic j direction is -3

  • You need to know how to find the component of a vector that is:

    • Parallel to another vector

    • Perpendicular to another vector

How do I find the component of a vector that is parallel to another vector?

  • Suppose you want to find the component of bold italic a that is acting in the direction of bold italic b

  • If you know the angle between bold italic a and bold italic b

    • The component of bold italic a acting parallel to bold italic b is open vertical bar bold italic a close vertical bar cos theta

  • You can also find the component using the scalar product

    • The formula is fraction numerator bold italic a times bold italic b over denominator open vertical bar bold italic b close vertical bar end fraction

How do I find the component of a vector that is perpendicular to another vector?

  • Suppose you want to find the component of bold italic a that is acting perpendicular to bold italic b

    • There are an infinite number of directions that are perpendicular to a vector in 3D

    • Therefore, you want to find the component that is in the plane formed by the two vectors

  • If you know the angle between bold italic a and bold italic b

    • The component of bold italic a acting perpendicular to bold italic b is open vertical bar bold italic a close vertical bar sin theta

  • You can also find the component using the vector product

    • The formula is fraction numerator open vertical bar bold italic a cross times bold italic b close vertical bar over denominator open vertical bar bold italic b close vertical bar end fraction 

Examiner Tips and Tricks

None of these formulas are given in the formula booklet. Therefore, you should draw a sketch to help you decide whether to use sine or cosine. And then you can use the given formulas for the scalar product and vector product to determine which one you need to use.

Vector diagram showing vector a split into perpendicular and parallel components to vector b, with trigonometric expressions for each component.
Example of the components of a vector which are parallel and perpendicular to another vector

Worked Example

A force with magnitude 10 N is acting on a bearing of 060° on an object which is moving with velocity vector v = 2i - 3j.

a) By finding the components of the force in the i and j direction, write down the force as a vector.

4uoe23xh_3-7-6-ib-ai-hl-components-of-vectors-we-sol-a

b) Find the component of the force acting parallel to the direction of the object.

ysOarxyw_3-7-6-ib-ai-hl-components-of-vectors-we-sol-b

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    Amber

    Author: Amber

    Expertise: Maths Content Creator

    Amber gained a first class degree in Mathematics & Meteorology from the University of Reading before training to become a teacher. She is passionate about teaching, having spent 8 years teaching GCSE and A Level Mathematics both in the UK and internationally. Amber loves creating bright and informative resources to help students reach their potential.

    Dan Finlay

    Reviewer: Dan Finlay

    Expertise: Maths Subject Lead

    Dan graduated from the University of Oxford with a First class degree in mathematics. As well as teaching maths for over 8 years, Dan has marked a range of exams for Edexcel, tutored students and taught A Level Accounting. Dan has a keen interest in statistics and probability and their real-life applications.