Given that the vector 𝐯 has components 0.6, negative 0.8, find the value of the magnitude of 𝐯.
In reading the question out, I gave away the fact that this notation at the end. The 𝐯 with the vertical bars on either side stands for the magnitude of the vector 𝐯. For a general vector with components 𝑥 and 𝑦, the magnitude of this vector is equal to the square root of 𝑥 squared plus 𝑦 squared.
If we think geometrically, then the magnitude of the vector is its length; it’s the distance between its initial and terminal points. And so this square root of 𝑥 squared plus 𝑦 squared comes from the Pythagorean theorem that being applied to the right triangle that we’ve drawn with the vector.
So what is the magnitude of our vector 𝐯? Well, we can substitute in the 𝑥- and 𝑦-components. So we get the square root of 0.6 squared plus negative 0.8 squared. And if we don’t rush straight to our calculators, we can use the fact that 0.6 squared is 0.36 and negative 0.8 squared is 0.64 to write this as the square root of 0.36 plus 0.64.
And we had to be slightly careful about our parentheses. Here, we have negative 0.8 squared and not negative 0.8 squared. And so we have 0.64 and not negative 0.64. Anyway, simplifying under the radical sign, we get the square root of one, which is of course just one. The magnitude of 𝐯, where 𝐯 has components 0.6 and negative 0.8, is one.