GCSE Maths Practice: vectors

Question 1 of 10

This question teaches how to calculate the magnitude (length) of a vector using Pythagoras' Theorem.

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\( \begin{array}{l}\text{Find the magnitude of the vector } \mathbf{v} = \begin{pmatrix}3\\4\end{pmatrix}.\end{array} \)

Choose one option:

Square each component, sum them, then take the square root.

The magnitude of a vector is its length, calculated as the distance from the origin (0,0) to the point represented by the vector. For a 2D vector \(\begin{pmatrix}x\\y\end{pmatrix}\), the magnitude is \(\sqrt{x^2+y^2}\). For example, for \(\begin{pmatrix}3\\4\end{pmatrix}\), calculate \(\sqrt{3^2+4^2} = \sqrt{9+16} = \sqrt{25} = 5\). Understanding magnitude is essential for physics applications like forces and displacement, as well as coordinate geometry. Practice with vectors in different quadrants, including negative components, helps build intuition for distance and direction. Visualizing the vector on a graph helps students see why Pythagoras’ Theorem applies: the x and y components form a right-angled triangle. Consistent practice ensures fluency in computing vector lengths for all GCSE problems and supports advanced topics in mechanics and vector algebra.