GCSE Maths Practice: vectors

Question 3 of 10

This question teaches combining scalar multiplication and vector subtraction.

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\( \begin{array}{l}\text{Find } 2\mathbf{a} - \mathbf{b} \text{ if } \mathbf{a} = \begin{pmatrix}1\\2\end{pmatrix} \text{ and } \mathbf{b} = \begin{pmatrix}3\\-1\end{pmatrix}.\end{array} \)

Choose one option:

Multiply by the scalar first, then subtract each component.

First, multiply vector \(\mathbf{a} = \begin{pmatrix}1\\2\end{pmatrix}\) by 2 to get \(2\mathbf{a} = \begin{pmatrix}2\\4\end{pmatrix}\). Then subtract \(\mathbf{b} = \begin{pmatrix}3\\-1\end{pmatrix}\) component-wise: top: 2-3=-1, bottom: 4-(-1)=5. The result is \(\begin{pmatrix}-1\\5\end{pmatrix}\). This demonstrates combining operations in vector algebra, which is essential in physics and coordinate geometry. Practice visualizing each step on a grid for clarity, and verify calculations component-wise to avoid mistakes. This method applies to more complex vector manipulations involving multiple scalars and vector operations.