Vector arithmetic
Objective. Add, subtract and scale column vectors.
- 1. Learn
- 2. Worked example
- 3. Practice
- 4. Feedback
- 5. Continue
Learn: the key idea
Vector arithmetic allows column vectors to be added, subtracted, or multiplied by a scalar number, each operation performed component by component. To add two vectors, add their top numbers together and their bottom numbers together separately; subtraction works the same way but subtracting corresponding components. To multiply a vector by a scalar, multiply both the top and bottom numbers by that scalar. For example, adding (2 over 3) and (−1 over 4) gives (2+(−1) over 3+4) = (1 over 7), while multiplying (2 over 3) by 3 gives (6 over 9).
Worked example
Given vectors a = (5 over −2) and b = (−3 over 6), calculate the vector 2a − b.
- 1First find 2a by doubling each component: 2a = (10 over −4).
- 2Now subtract b component by component: top is 10 − (−3) = 13.
- 3Bottom is −4 − 6 = −10.
- 4Combine the results: 2a − b = (13 over −10).
Answer: (13 over −10)
Concept mastery
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Practice
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Start practisingCommon mistake. Pupils sometimes add the top number of one vector to the bottom number of the other by mistake, mixing up the components instead of keeping horizontal with horizontal and vertical with vertical. Always add or subtract matching components: top with top, bottom with bottom.
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