In Exercises 39–46, find the unit vector that has the same direction as the vector v.
v = -5j

Verified step by step guidance

1. Identify the given vector \( \mathbf{v} = -5\mathbf{j} \). This means the vector points in the negative y-direction with a magnitude of 5.
2. Recall that the unit vector \( \mathbf{u} \) in the direction of \( \mathbf{v} \) is found by dividing \( \mathbf{v} \) by its magnitude \( \|\mathbf{v}\| \). The formula is: \[ \mathbf{u} = \frac{\mathbf{v}}{\|\mathbf{v}\|} \]
3. Calculate the magnitude of \( \mathbf{v} \). Since \( \mathbf{v} = 0\mathbf{i} - 5\mathbf{j} \), the magnitude is: \[ \|\mathbf{v}\| = \sqrt{0^2 + (-5)^2} = \sqrt{25} \]
4. Divide each component of \( \mathbf{v} \) by the magnitude \( \|\mathbf{v}\| \) to get the unit vector components: \[ \mathbf{u} = \left( \frac{0}{\|\mathbf{v}\|}, \frac{-5}{\|\mathbf{v}\|} \right) \]
5. Write the unit vector \( \mathbf{u} \) in terms of the unit vectors \( \mathbf{i} \) and \( \mathbf{j} \) using the components found in the previous step.

Key Concepts

Vector Direction

The direction of a vector is the orientation in space it points to, regardless of its magnitude. Two vectors have the same direction if one is a scalar multiple of the other. Understanding direction helps in finding unit vectors that preserve this orientation.

Magnitude of a Vector

The magnitude (or length) of a vector is the distance from the origin to the point represented by the vector. It is calculated using the Pythagorean theorem for components. Magnitude is essential for normalizing a vector to find its unit vector.

Unit Vector

A unit vector has a magnitude of exactly one and points in a specific direction. To find a unit vector in the same direction as a given vector, divide the vector by its magnitude. This process normalizes the vector, preserving direction but standardizing length.