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


v = 4i - 2j

Verified step by step guidance

1. Identify the given vector \( \mathbf{v} = 4\mathbf{i} - 2\mathbf{j} \). This means the vector has components \( (4, -2) \).
2. Calculate the magnitude (length) of the vector \( \mathbf{v} \) using the formula: \[ \|\mathbf{v}\| = \sqrt{(4)^2 + (-2)^2} \]
3. Simplify the expression under the square root to find the magnitude: \[ \|\mathbf{v}\| = \sqrt{16 + 4} \]
4. Find the unit vector \( \mathbf{u} \) in the same direction as \( \mathbf{v} \) by dividing each component of \( \mathbf{v} \) by its magnitude: \[ \mathbf{u} = \left( \frac{4}{\|\mathbf{v}\|}, \frac{-2}{\|\mathbf{v}\|} \right) \]
5. Express the unit vector in terms of the standard unit vectors \( \mathbf{i} \) and \( \mathbf{j} \): \[ \mathbf{u} = \frac{4}{\|\mathbf{v}\|} \mathbf{i} - \frac{2}{\|\mathbf{v}\|} \mathbf{j} \]

Key Concepts

Vector Components

A vector in two dimensions can be expressed in terms of its components along the x and y axes, typically written as v = ai + bj, where a and b are scalar values. Understanding these components is essential for operations like finding magnitude and direction.

Magnitude of a Vector

The magnitude (or length) of a vector v = ai + bj is calculated using the Pythagorean theorem as √(a² + b²). This scalar value represents the distance from the origin to the point defined by the vector components.

Unit Vector

A unit vector has a magnitude of 1 and points in the same direction as the original vector. It is found by dividing each component of the vector by its magnitude, effectively normalizing the vector without changing its direction.