In Exercises 39–46, find the unit vector that has the same direction as the vector v.v = 6i

Verified step by step guidance

1. Identify the given vector \( \mathbf{v} = 6\mathbf{i} \).
2. Calculate the magnitude of the vector \( \mathbf{v} \). The magnitude \( ||\mathbf{v}|| \) is given by \( ||\mathbf{v}|| = \sqrt{6^2} \).
3. Simplify the expression for the magnitude: \( ||\mathbf{v}|| = \sqrt{36} \).
4. Determine the unit vector \( \mathbf{u} \) by dividing the vector \( \mathbf{v} \) by its magnitude: \( \mathbf{u} = \frac{\mathbf{v}}{||\mathbf{v}||} = \frac{6\mathbf{i}}{\sqrt{36}} \).
5. Simplify the expression for the unit vector: \( \mathbf{u} = \frac{6\mathbf{i}}{6} \).

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 having a standardized length.

Magnitude of a Vector

The magnitude of a vector is a measure of its length and is calculated using the formula √(x² + y² + z²) for a vector in three-dimensional space. For a two-dimensional vector, it simplifies to √(x² + y²). Understanding how to compute the magnitude is essential for normalizing a vector to find its unit vector.

Vector Notation

Vector notation typically uses letters with an arrow or bold type to represent vectors, such as v = 6i, where 'i' denotes the unit vector in the x-direction. Recognizing this notation is crucial for interpreting vector components and performing operations like finding unit vectors, as it helps in visualizing the vector's direction and magnitude.