Find the magnitude of the vector 𝐯 shown on the grid of unit squares below.
Any vector in two dimensions can be written in terms of its 𝑥- and 𝑦-components as shown. The 𝑥-component is the horizontal displacement and the 𝑦-component, the vertical displacement. We assume the positive direction of the 𝑥-component is to the right, and the positive direction for the 𝑦-component is upwards. In this question, vector 𝐯 is moving left and downwards. This means that both components will be negative. To get from the initial to the terminal point of the vector, we move three units left and two units down. This means that vector 𝐯 has components negative three and negative two.
We are asked to calculate the magnitude of the vector. We know this is equal to the square root of 𝑥 squared plus 𝑦 squared. The magnitude of vector 𝐯 is therefore equal to the square root of negative three squared plus negative two squared. Squaring a negative number gives a positive answer, which means that negative three squared is equal to nine and negative two squared is equal to four. The magnitude of vector 𝐯 is equal to the square root of 13 or root 13.
We could also have calculated this by looking at the diagram and using the Pythagorean theorem. The magnitude of vector 𝐯 will be equal to the hypotenuse of our right triangle. The Pythagorean theorem states that 𝑐 squared is equal to 𝑎 squared plus 𝑏 squared, where 𝑐 is the length of the hypotenuse or longest side of the triangle. Substituting in our values gives us eight squared is equal to two squared plus three squared. The right-hand side simplifies to give us 13. And square rooting both sides of the equation gives us ℎ is equal to root 13. The length of the hypotenuse of the right triangle is root 13, which confirms that the magnitude of vector 𝐯 is also equal to root 13.