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


v = i - j

Verified step by step guidance

1. Calculate the magnitude of the vector \( \mathbf{v} = \mathbf{i} - \mathbf{j} \) using the formula \( \| \mathbf{v} \| = \sqrt{a^2 + b^2} \), where \( a \) and \( b \) are the components of the vector.
2. Substitute the components of \( \mathbf{v} \) into the magnitude formula: \( a = 1 \) and \( b = -1 \).
3. Simplify the expression to find the magnitude: \( \| \mathbf{v} \| = \sqrt{1^2 + (-1)^2} \).
4. Divide each component of the vector \( \mathbf{v} \) by its magnitude to find the unit vector: \( \mathbf{u} = \frac{1}{\| \mathbf{v} \|} (\mathbf{i} - \mathbf{j}) \).
5. Express the unit vector in terms of its components: \( \mathbf{u} = \left( \frac{1}{\| \mathbf{v} \|}, \frac{-1}{\| \mathbf{v} \|} \right) \).

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, ensuring it retains its direction while having a length of one.

Magnitude of a Vector

The magnitude of a vector is a measure of its length and is calculated using the formula √(x² + y²) for a two-dimensional vector represented as (x, y). In the case of the vector v = i - j, the components are 1 and -1, leading to a magnitude of √(1² + (-1)²) = √2, which is essential for normalizing the vector.

Direction of a Vector

The direction of a vector is determined by the angle it makes with a reference axis, typically the x-axis. For the vector v = i - j, the direction can be visualized in the Cartesian plane, where the vector points diagonally downwards. Understanding direction is crucial when finding a unit vector, as it ensures the resulting unit vector points in the same way as the original vector.