Explainer: Magnitude of a 2D Vector

In this explainer, we will learn how to determine the magnitude of two-dimensional vectors.

Recall that a vector has two aspects: direction and magnitude. The magnitude is the vector’s “size ” or “length. ”

Example 1: Visually Comparing the Magnitude of Vectors

Which vector has the greatest magnitude?

Answer

Visual inspection tells us that c is the longest, so it has the greatest magnitude. Note that we can move c to have the same initial point as a and then rotate it so that the 2×5 grid it lies in matches the 2×3 grid of a. This makes its greater length clear.

We denote the magnitude of the vector v by ||v and use Pythagoras’ theorem to compute it, against the grid of unit squares.

Example 2: Finding the Magnitude of a Vector on a Coordinate Grid

The vector v is shown on the grid of unit squares below. Find the value of ||v.

Answer

The magnitude (or length) of the vector is 4+3=16+9=25=5.

The magnitude is calculated in the same way from its components.

Example 3: Finding the Magnitude of a Vector from Its Initial Point and Endpoint

What is the magnitude of the vector 𝐴𝐵, where A=6,9 and B=9,1?

Answer

We have that 𝐴𝐵=𝑎,𝑏,𝑎=9(6)=15,𝑏=1(9)=8,||𝐴𝐵||=𝑎+𝑏=15+8=225+64=289=17.wheresothemagnitude

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