# Video: AQA GCSE Mathematics Foundation Tier Pack 4 • Paper 2 • Question 26

𝐚 and 𝐛 are column vectors. Where 𝐚 = (3, −7), 𝐛 = (−2, 1). Calculate 2𝐛 − 5𝐚.

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### Video Transcript

𝐚 and 𝐛 are column vectors. Where 𝐚 equals three, negative seven and 𝐛 equals negative two, one. Calculate two 𝐛 minus five 𝐚.

When multiplying any vector by a constant we need to multiply both the 𝑥- and 𝑦-values. For example, multiplying the vector 𝑥, 𝑦 by the constant 𝑐 is equal to the vector 𝑐𝑥, 𝑐𝑦. Let’s firstly calculate the vectors two 𝐛 and five 𝐚.

The vector two 𝐛 is equal to two multiplied by the vector negative two, one. Two multiplied by negative two is negative four. And two multiplied by one is equal to two. Therefore, the vector two 𝐛 is equal to negative four, two.

To calculate the vector five 𝐚, we multiply the constant five by the vector three, negative seven. Five multiplied by three is equal to 15. And five multiplied by negative seven is equal to negative 35. Therefore, the vector five 𝐚 is equal to 15, negative 35.

We need to calculate two 𝐛 minus five 𝐚. We need to subtract the vector 15, negative 35 from the vector negative four, two. In order to do this, we subtract the top numbers, or the 𝑥-values, and separately we subtract the bottom numbers, or 𝑦-values.

Negative four minus 15 is equal to negative 19. When we are subtracting from a negative number, it moves further away from zero. When we are subtracting a negative number, or two negative signs are touching, it becomes a positive. Two minus negative 35 is equal to two plus 35. This is equal to 37. The vector two 𝐛 minus five 𝐚 is equal to the column vector negative 19, 37.