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


v = 8i - 6j

Verified step by step guidance

1. Calculate the magnitude of the vector \( \mathbf{v} = 8\mathbf{i} - 6\mathbf{j} \) using the formula \( ||\mathbf{v}|| = \sqrt{a^2 + b^2} \), where \( a = 8 \) and \( b = -6 \).
2. Substitute the values into the formula: \( ||\mathbf{v}|| = \sqrt{8^2 + (-6)^2} \).
3. Simplify the expression under the square root: \( ||\mathbf{v}|| = \sqrt{64 + 36} \).
4. Calculate the square root to find the magnitude of \( \mathbf{v} \).
5. Divide each component of the vector \( \mathbf{v} \) by its magnitude to find the unit vector: \( \mathbf{u} = \frac{1}{||\mathbf{v}||}(8\mathbf{i} - 6\mathbf{j}) \).

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, preserving its direction while scaling its length to one.

Magnitude of a Vector

The magnitude of a vector is a measure of its length in space, calculated using the formula √(x² + y²) for a two-dimensional vector. For the vector v = 8i - 6j, the magnitude is √(8² + (-6)²) = √(64 + 36) = √100 = 10. This value is essential for normalizing the vector to find the unit vector.

Vector Components

Vectors in two dimensions can be expressed in terms of their components along the x-axis and y-axis, typically denoted as ai + bj. In the vector v = 8i - 6j, '8' is the x-component and '-6' is the y-component. Understanding these components is crucial for calculating the magnitude and subsequently finding the unit vector.