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. Step 1: Understand the problem. We need to find a unit vector that has the same direction as the given vector \( \mathbf{v} = \mathbf{i} + \mathbf{j} \).
2. Step 2: Recall that a unit vector is a vector with a magnitude of 1. To find the unit vector in the direction of \( \mathbf{v} \), we need to divide \( \mathbf{v} \) by its magnitude.
3. Step 3: Calculate the magnitude of \( \mathbf{v} \). The magnitude of a vector \( \mathbf{v} = a\mathbf{i} + b\mathbf{j} \) is given by \( \|\mathbf{v}\| = \sqrt{a^2 + b^2} \). For \( \mathbf{v} = \mathbf{i} + \mathbf{j} \), this becomes \( \|\mathbf{v}\| = \sqrt{1^2 + 1^2} = \sqrt{2} \).
4. Step 4: Divide each component of \( \mathbf{v} \) by its magnitude to find the unit vector. The unit vector \( \mathbf{u} \) is given by \( \mathbf{u} = \frac{1}{\|\mathbf{v}\|}(\mathbf{i} + \mathbf{j}) = \frac{1}{\sqrt{2}}\mathbf{i} + \frac{1}{\sqrt{2}}\mathbf{j} \).
5. Step 5: Verify that the resulting vector is indeed a unit vector by checking its magnitude. The magnitude of \( \mathbf{u} \) should be 1. Calculate \( \|\mathbf{u}\| = \sqrt{\left(\frac{1}{\sqrt{2}}\right)^2 + \left(\frac{1}{\sqrt{2}}\right)^2} = \sqrt{\frac{1}{2} + \frac{1}{2}} = \sqrt{1} = 1 \).

Key Concepts

Unit Vector

A unit vector is a vector that has a magnitude of one and indicates direction. To convert any vector into a unit vector, you divide the vector by its magnitude. This process ensures that the resulting vector maintains the same direction as the original but has a standardized length of one.

Magnitude of a Vector

The magnitude of a vector is a measure of its length in space, calculated using the Pythagorean theorem. For a vector represented as v = ai + bj, the magnitude is given by √(a² + b²). Understanding how to compute the magnitude is essential for finding the unit vector, as it serves as the denominator in the conversion process.

Direction of a Vector

The direction of a vector indicates the path along which it points in space. In a two-dimensional Cartesian coordinate system, the direction can be represented by the angle the vector makes with the positive x-axis. When finding a unit vector, it is crucial to maintain the original vector's direction while adjusting its magnitude to one.