Unit Vectors and Directions in Trigonometry

Unit Vectors and Directions

Definition of a Unit Vector

A unit vector is a vector with a magnitude (length) of 1. It indicates direction only and is commonly used to specify directions in space.

• Notation: A unit vector in the direction of vector v is often written as u or ̂v.

• Formula: To find the unit vector in the direction of a given vector v, divide the vector by its magnitude:

$\mathbf{u} = \frac{\mathbf{v}}{\|\mathbf{v}\|}$

• Where $\|\mathbf{v}\|$ is the magnitude (length) of vector $\mathbf{v}$.

Finding the Magnitude of a Vector

The magnitude of a vector $\mathbf{v} = \langle a, b \rangle$ in two dimensions is given by:

$\|\mathbf{v}\| = \sqrt{a^2 + b^2}$

For a three-dimensional vector $\mathbf{v} = \langle a, b, c \rangle$:

$\|\mathbf{v}\| = \sqrt{a^2 + b^2 + c^2}$

Example: Finding a Unit Vector

Given $\mathbf{v} = \langle 3, 4 \rangle$:

• Find the magnitude:

$\|\mathbf{v}\| = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5$

• Divide each component by the magnitude:

$\mathbf{u} = \frac{1}{5}\langle 3, 4 \rangle = \langle \frac{3}{5}, \frac{4}{5} \rangle$

Therefore, the unit vector in the direction of $\langle 3, 4 \rangle$ is $\langle \frac{3}{5}, \frac{4}{5} \rangle$.

Practice Problems

• Find the unit vector in the direction of $\mathbf{v} = \langle 4, 10 \rangle$.

• Find the unit vector in the direction of $\mathbf{v} = \langle 12, -5 \rangle$.

Solution Steps:

1. Calculate the magnitude $\|\mathbf{v}\|$.

2. Divide each component of $\mathbf{v}$ by $\|\mathbf{v}\|$.

Applications

• Unit vectors are used to describe directions in physics, engineering, and navigation.

• They are essential in expressing vectors in terms of their direction only, independent of magnitude.