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 \( \| \mathbf{v} \| = \sqrt{2} \).
4. Find the unit vector \( \mathbf{u} \) in the same direction as \( \mathbf{v} \) by dividing each component of \( \mathbf{v} \) by its magnitude: \( \mathbf{u} = \frac{1}{\| \mathbf{v} \|} \mathbf{v} = \left\langle \frac{1}{\sqrt{2}}, \frac{1}{\sqrt{2}} \right\rangle \).
5. Express the unit vector in terms of \( \mathbf{i} \) and \( \mathbf{j} \) as \( \mathbf{u} = \frac{1}{\sqrt{2}} \mathbf{i} + \frac{1}{\sqrt{2}} \mathbf{j} \).

Key Concepts

Vector Components and Notation

A vector in two dimensions can be expressed using unit vectors i and j, representing the x and y components respectively. For example, v = i + j means the vector has components (1, 1) along the x and y axes.

Magnitude of a Vector

The magnitude (or length) of a vector v = (x, y) is found using the Pythagorean theorem: |v| = √(x² + y²). This scalar value represents the distance from the origin to the point (x, y) in the plane.

Unit Vector in the Direction of a Given 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, resulting in a vector of length one.