Problem 46 - Precalculus with Limits

Question

In Exercises 39-48, find a unit vector in the direction of the given vector. Verify that the result has a magnitude of 1. $\mathbf{w} = -6\mathbf{i}

Step-by-Step Solution

Answer

The unit vector in the direction of the given vector \(\mathbf{w}\) is \(-\mathbf{i}\).

Step 1: Calculate the magnitude of the vector

The magnitude (or length) of vector \(\mathbf{w}\) is found using the formula for Euclidean norm. For this vector, that is \(\|\mathbf{w}\| = \sqrt{(-6)^2} = 6\).

Step 2: Find the unit vector

The unit vector is found by dividing vector \(\mathbf{w}\) by its magnitude. So, in this example that would be \(\mathbf{u}_w = \frac{\mathbf{w}}{\|\mathbf{w}\|} = \frac{-6\mathbf{i}}{6} = -\mathbf{i}\)

Step 3: Verify the magnitude of the unit vector

Now we verify that the magnitude of the unit vector is 1 by calculating the Euclidean length. For our unit vector, the magnitude is \(\|\mathbf{u}_w\| = \sqrt{(-1)^2} = 1\)