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


v = 4i - 2j

Verified step by step guidance

1. Calculate the magnitude of the vector $\mathbf{v}=4\mathbf{i}-2\mathbf{j}$ using the formula $\Vert \mathbf{v}\Vert =\sqrt{a^2+b^2}$, where $a=4$ and $b=-2$.
2. Substitute the values into the formula: $\Vert \mathbf{v}\Vert =\sqrt{4^2+(-2{)}^2}$.
3. Simplify the expression under the square root: $\Vert \mathbf{v}\Vert =\sqrt{16+4}$.
4. Calculate the square root: $\Vert \mathbf{v}\Vert =\sqrt{20}$.
5. Divide each component of the vector $\mathbf{v}$ by its magnitude to find the unit vector: $\mathbf{u}=\frac{1}{\Vert \mathbf{v}\Vert }(4\mathbf{i}-2\mathbf{j})$.

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 2D vector represented as v = xi + yj. For the vector v = 4i - 2j, the magnitude is √(4² + (-2)²) = √(16 + 4) = √20, which is essential for determining the unit vector.

Vector Components

Vectors in two-dimensional space can be expressed in terms of their components along the x-axis and y-axis, typically written as v = xi + yj. In the vector v = 4i - 2j, the components are 4 (along the x-axis) and -2 (along the y-axis). Understanding these components is crucial for calculating both the magnitude and the unit vector.