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. Identify the given vector \( \mathbf{v} = \mathbf{i} - \mathbf{j} \), which can be written in component form as \( \mathbf{v} = \langle 1, -1 \rangle \).
2. Calculate the magnitude (length) of the vector \( \mathbf{v} \) using the formula \( \| \mathbf{v} \| = \sqrt{v_x^2 + v_y^2} \). Substitute the components to get \( \| \mathbf{v} \| = \sqrt{1^2 + (-1)^2} \).
3. Simplify the expression under the square root to find the magnitude \( \| \mathbf{v} \| \).
4. Find the unit vector \( \mathbf{u} \) in the same direction as \( \mathbf{v} \) by dividing each component of \( \mathbf{v} \) by its magnitude: \( \mathbf{u} = \frac{1}{\| \mathbf{v} \|} \mathbf{v} = \left\langle \frac{1}{\| \mathbf{v} \|}, \frac{-1}{\| \mathbf{v} \|} \right\rangle \).
5. Express the unit vector \( \mathbf{u} \) in terms of \( \mathbf{i} \) and \( \mathbf{j} \) as \( \mathbf{u} = \frac{1}{\| \mathbf{v} \|} \mathbf{i} - \frac{1}{\| \mathbf{v} \|} \mathbf{j} \).

Key Concepts

Vector Direction

The direction of a vector is the orientation it points to in space, independent of its length. Two vectors have the same direction if one is a scalar multiple of the other. Understanding direction is essential to find a unit vector that aligns with the original vector.

Vector Magnitude (Length)

The magnitude of a vector is its length, calculated using the square root of the sum of the squares of its components. For vector v = i - j, the magnitude is √(1² + (-1)²) = √2. This value is used to normalize the vector.

Unit Vector

A unit vector has a magnitude of 1 and points in a specific direction. To find a unit vector in the same direction as v, divide each component of v by its magnitude. This process scales the vector to length one without changing its direction.