### Video Transcript

True or False: The magnitude of a 3D vector represents the length of the vector.

In order to answer this question, letβs remind ourselves what we mean when we talk about the magnitude of a two-dimensional vector. Suppose we have a two-dimensional vector denoted π. And itβs given by π₯π’ plus π¦π£, where π’ and π£ are perpendicular unit vectors. We can then represent the vector π assuming π₯ and π¦ are positive as shown. We then see that the lengths represented by the distances traveled in the π’- and π£-directions are perpendicular. So we have a right triangle.

We denote the length of the actual vector π 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 π. Itβs the magnitude of π squared equals π₯ squared plus π¦ squared, which in turn means that the magnitude of π is the square root of π₯ squared plus π¦ squared. Letβs now suppose we have a three-dimensional vector π, which is given by π₯π’ plus π¦π£ plus π§π€. We still denote the magnitude of π using these horizontal bars. Then, weβre able to extend the idea of the Pythagorean theorem in two dimensions to find the magnitude of π. Itβs the square root of π₯ squared plus π¦ squared plus π§ squared. In the same way as in two dimensions then, the magnitude of π 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 π₯π’ plus π¦π£ plus π§π€, its magnitude is the square root of π₯ squared plus π¦ squared plus π§ squared.