Questions: Find u for the given vector. u = [4, 3, 1] Give a unit vector in the direction of u.

Transcript text: Find $\|\mathbf{u}\|$ for the given vector. \[ \mathbf{u}=\left[\begin{array}{l} 4 \\ 3 \\ 1 \end{array}\right] \] Give a unit vector in the direction of $\mathbf{u}$.

Solution

Solution Steps

Step 1: Calculate the Magnitude of the Vector $\mathbf{u}$

The magnitude (or norm) of a vector $\mathbf{u} = \begin{bmatrix} 4 \\ 3 \\ 1 \end{bmatrix}$ is calculated using the formula:

\[ \|\mathbf{u}\| = \sqrt{u_1^2 + u_2^2 + u_3^2} \]

Substitute the components of $\mathbf{u}$:

\[ \|\mathbf{u}\| = \sqrt{4^2 + 3^2 + 1^2} = \sqrt{16 + 9 + 1} = \sqrt{26} \]

Step 2: Find the Unit Vector in the Direction of $\mathbf{u}$

A unit vector $\mathbf{v}$ in the direction of $\mathbf{u}$ is given by:

\[ \mathbf{v} = \frac{\mathbf{u}}{\|\mathbf{u}\|} \]

Substitute the vector $\mathbf{u}$ and its magnitude:

\[ \mathbf{v} = \frac{1}{\sqrt{26}} \begin{bmatrix} 4 \\ 3 \\ 1 \end{bmatrix} = \begin{bmatrix} \frac{4}{\sqrt{26}} \\ \frac{3}{\sqrt{26}} \\ \frac{1}{\sqrt{26}} \end{bmatrix} \]