True or False: The magnitude of a 3D vector represents the length of the vector.
In order to answer this question, let’s remind ourselves what we mean when we talk about the magnitude of a two-dimensional vector. Suppose we have a two-dimensional vector denoted 𝐚. And it’s given by 𝑥𝐢 plus 𝑦𝐣, where 𝐢 and 𝐣 are perpendicular unit vectors. We can then represent the vector 𝐚 assuming 𝑥 and 𝑦 are positive as shown. We then see that the lengths represented by the distances traveled in the 𝐢- and 𝐣-directions are perpendicular. So we have a right triangle.
We denote the length of the actual vector 𝐚 using these two bars, and we call it its magnitude. Since the triangle is right-angled, we can use the Pythagorean theorem to find the magnitude of 𝐚. It’s the magnitude of 𝐚 squared equals 𝑥 squared plus 𝑦 squared, which in turn means that the magnitude of 𝐚 is the square root of 𝑥 squared plus 𝑦 squared. Let’s now suppose we have a three-dimensional vector 𝐛, which is given by 𝑥𝐢 plus 𝑦𝐣 plus 𝑧𝐤. We still denote the magnitude of 𝐛 using these horizontal bars. Then, we’re able to extend the idea of the Pythagorean theorem in two dimensions to find the magnitude of 𝐛. It’s the square root of 𝑥 squared plus 𝑦 squared plus 𝑧 squared. In the same way as in two dimensions then, the magnitude of 𝐛 represents the length of the vector itself.
And so the answer is true. The magnitude of a 3D vector represents the length of the vector. Further, if we call the vector 𝑥𝐢 plus 𝑦𝐣 plus 𝑧𝐤, its magnitude is the square root of 𝑥 squared plus 𝑦 squared plus 𝑧 squared.