Find the unit vector in the direction of \( \mathbf{v} \). \[ \mathbf{v}=-7.1 \mathbf{i}+3.5 \mathbf{j} \]

To find the **unit vector** in the direction of the vector \( \mathbf{v} = -7.1\,\mathbf{i} + 3.5\,\mathbf{j} \), follow these steps: ### 1. Calculate the Magnitude of \( \mathbf{v} \): The magnitude (or length) of \( \mathbf{v} \), denoted \( \| \mathbf{v} \| \), is given by: \[ \| \mathbf{v} \| = \sqrt{(-7.1)^2 + (3.5)^2} \] Compute the squares: \[ (-7.1)^2 = 50.41 \quad \text{and} \quad (3.5)^2 = 12.25 \] Sum them up: \[ \| \mathbf{v} \| = \sqrt{50.41 + 12.25} = \sqrt{62.66} \approx 7.916 \] ### 2. Divide Each Component by the Magnitude: A **unit vector** \( \mathbf{u} \) in the direction of \( \mathbf{v} \) is given by: \[ \mathbf{u} = \frac{\mathbf{v}}{\| \mathbf{v} \|} = \left( \frac{-7.1}{7.916} \right)\mathbf{i} + \left( \frac{3.5}{7.916} \right)\mathbf{j} \] Calculate each component: \[ \frac{-7.1}{7.916} \approx -0.897 \quad \text{and} \quad \frac{3.5}{7.916} \approx 0.442 \] ### 3. Write the Unit Vector: \[ \mathbf{u} \approx -0.897\,\mathbf{i} + 0.442\,\mathbf{j} \] ### **Final Answer:** A unit vector in the direction of **v** is –0.897 i plus 0.442 j. In symbols, u = −0.897 i + 0.442 j