In Exercises 31–32, find the unit vector that has the same direction as the vector v. v = -i + 2j

Verified step by step guidance

1. <insert step 1: Understand that a unit vector is a vector with a magnitude of 1 that points in the same direction as the given vector.>
2. <insert step 2: Calculate the magnitude of the vector \( \mathbf{v} = -\mathbf{i} + 2\mathbf{j} \) using the formula \( \|\mathbf{v}\| = \sqrt{(-1)^2 + (2)^2} \).>
3. <insert step 3: Simplify the expression to find the magnitude \( \|\mathbf{v}\| = \sqrt{1 + 4} = \sqrt{5} \).>
4. <insert step 4: Divide each component of the vector \( \mathbf{v} \) by its magnitude to find the unit vector: \( \mathbf{u} = \left( \frac{-1}{\sqrt{5}} \right)\mathbf{i} + \left( \frac{2}{\sqrt{5}} \right)\mathbf{j} \).>
5. <insert step 5: Recognize that the unit vector \( \mathbf{u} \) maintains the direction of \( \mathbf{v} \) but has a magnitude of 1.>

Key Concepts

Unit Vector

A unit vector is a vector that has a magnitude of one and indicates direction. To find a unit vector in the same direction as a given vector, you divide the vector by its magnitude. This process normalizes the vector, allowing it to retain its direction while simplifying its length to one.

Magnitude of a Vector

The magnitude of a vector is a measure of its length and is calculated using the formula √(x² + y²) for a two-dimensional vector represented as (x, y). In the context of the vector v = -i + 2j, the magnitude helps determine how to scale the vector down to a unit vector, ensuring the direction remains unchanged while the length is adjusted.

Direction of a Vector

The direction of a vector is defined by the angle it makes with a reference axis, typically the x-axis. In trigonometry, the direction can also be expressed using unit vectors, which indicate the same orientation as the original vector. Understanding direction is crucial when finding a unit vector, as it ensures that the resulting vector points in the same way as the original.