Questions: For any nonzero vector v, the unit vector that has the same direction as v is . To find this vector, divide v by its magnitude.

Transcript text: For any nonzero vector $\mathbf{v}$, the unit vector that has the same direction as $\mathbf{v}$ is $\square$ To find this vector, divide $\mathbf{v}$ by its $\square$ magnitude.

Solution

Solution Steps

Solution Approach

To find the unit vector in the same direction as a given nonzero vector $\mathbf{v}$, you need to divide the vector by its magnitude. The magnitude of a vector $\mathbf{v} = (v_1, v_2, \ldots, v_n)$ is calculated using the formula $\sqrt{v_1^2 + v_2^2 + \ldots + v_n^2}$. Once you have the magnitude, divide each component of the vector by this magnitude to get the unit vector.

Step 1: Calculate the Magnitude of the Vector

Given the vector $\mathbf{v} = [3, 4]$, we first calculate its magnitude using the formula:

\[ \|\mathbf{v}\| = \sqrt{v_1^2 + v_2^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]

Step 2: Find the Unit Vector

To find the unit vector $\mathbf{u}$ in the same direction as $\mathbf{v}$, we divide each component of $\mathbf{v}$ by its magnitude:

\[ \mathbf{u} = \frac{\mathbf{v}}{\|\mathbf{v}\|} = \left[\frac{3}{5}, \frac{4}{5}\right] = [0.6, 0.8] \]