In Exercises 39–46, find the unit vector that has the same direction as the vector v.
v = 3i - 4j

Verified step by step guidance

1. Identify the given vector \( \mathbf{v} = 3\mathbf{i} - 4\mathbf{j} \). This means the vector components are \( v_x = 3 \) and \( v_y = -4 \).
2. Calculate the magnitude (length) of the vector \( \mathbf{v} \) using the formula: \[ \\text{magnitude} = ||\mathbf{v}|| = \sqrt{v_x^2 + v_y^2} \] Substitute the components to get: \[ ||\mathbf{v}|| = \sqrt{3^2 + (-4)^2} \]
3. Simplify the expression inside the square root to find the magnitude, but do not calculate the final numeric value yet.
4. To find the unit vector \( \mathbf{u} \) in the same direction as \( \mathbf{v} \), divide each component of \( \mathbf{v} \) by its magnitude: \[ \mathbf{u} = \left( \frac{v_x}{||\mathbf{v}||}, \frac{v_y}{||\mathbf{v}||} \right) \]
5. Write the unit vector explicitly as: \[ \mathbf{u} = \frac{1}{||\mathbf{v}||} (3\mathbf{i} - 4\mathbf{j}) \] This expresses the unit vector in the same direction as \( \mathbf{v} \).

Key Concepts

Vector Components and Notation

A vector in two dimensions is expressed in terms of its components along the x and y axes, often written as v = ai + bj, where i and j are unit vectors in the x and y directions. Understanding this notation helps in identifying the vector's direction and magnitude.

Magnitude of a Vector

The magnitude (or length) of a vector v = ai + bj is found using the Pythagorean theorem: |v| = √(a² + b²). This scalar value represents the distance from the origin to the point (a, b) and is essential for normalizing the vector.

Unit Vector and Normalization

A unit vector has a magnitude of 1 and points in the same direction as the original vector. To find it, divide each component of the vector by its magnitude, resulting in a vector of length one that preserves the original direction.