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

To find the unit vector \( \mathbf{u} \) in the direction of \( \mathbf{v} \), we first calculate the magnitude of \( \mathbf{v} \): \[ |\mathbf{v}| = \sqrt{(-7.1)^2 + (3.5)^2} = \sqrt{50.41 + 12.25} = \sqrt{62.66} \approx 7.93. \] Now, we divide \( \mathbf{v} \) by its magnitude: \[ \mathbf{u} = \frac{\mathbf{v}}{|\mathbf{v}|} = \frac{-7.1 \mathbf{i} + 3.5 \mathbf{j}}{7.93} \approx -0.894 \mathbf{i} + 0.441 \mathbf{j}. \] So, the unit vector \( \mathbf{u} \) is approximately \[ \mathbf{u} \approx -0.894 \mathbf{i} + 0.441 \mathbf{j}. \] --- Did you know that unit vectors are often used in physics to represent directional quantities without concern for magnitude? They simplify many calculations, especially when dealing with forces or velocity in multiple dimensions. Instead of calculating the whole vector each time, you just scale it by the unit vector! In real-life applications, navigating using GPS coordinates can be thought of in terms of unit vectors. Each coordinate represents a direction in a two-dimensional space. By breaking down complex movements into simple unit vectors, pilots, drivers, and delivery drones can efficiently manage their paths, helping to avoid obstacles while staying on course.