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. Identify the given vector \( \mathbf{v} = -\mathbf{i} + 2\mathbf{j} \), which can be written in component form as \( \mathbf{v} = (-1, 2) \).
2. Calculate the magnitude (length) of the vector \( \mathbf{v} \) using the formula: \[ \\text{magnitude} = ||\mathbf{v}|| = \\sqrt{(-1)^2 + 2^2} \]
3. Simplify the expression under the square root to find the magnitude: \[ ||\mathbf{v}|| = \\sqrt{1 + 4} \]
4. Find the unit vector \( \mathbf{u} \) by dividing each component of \( \mathbf{v} \) by its magnitude: \[ \mathbf{u} = \left( \frac{-1}{||\mathbf{v}||}, \frac{2}{||\mathbf{v}||} \right) \]
5. Express the unit vector in terms of \( \mathbf{i} \) and \( \mathbf{j} \) as: \[ \mathbf{u} = \frac{-1}{||\mathbf{v}||} \mathbf{i} + \frac{2}{||\mathbf{v}||} \mathbf{j} \]

Key Concepts

Vector Direction

The direction of a vector is the orientation it points to in space, independent of its length. Two vectors have the same direction if one is a scalar multiple of the other. Understanding direction helps in finding a unit vector that points the same way as the original vector.

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 as the square root of the sum of the squares of its components. Magnitude is essential for normalizing a vector to unit length.

Unit Vector

A unit vector has a magnitude of exactly one and indicates direction only. To find a unit vector in the same direction as a given vector, divide each component of the vector by its magnitude. This process is called normalization and preserves direction while standardizing length.