Learning Outcomes
- Evaluate trigonometric functions using the unit circle
- Find a unit vector in the direction of v
In the Directional Derivatives and the Gradient section, we will learn how to further apply partial derivatives to directional derivatives and the gradient. Here we will review how to evaluate trigonometric functions and find a unit vector.
Evaluate Trigonometric Functions Using the Unit Circle
(See Module 4, Skills Review for Functions of Several Variables and Limits and Continuity)
Finding a Unit Vector
We call a vector with a magnitude of 1 a unit vector. This allows us to preserve the direction of the original vector while simplifying calculations.
Unit vectors are defined in terms of components. The horizontal unit vector is written as [latex]\boldsymbol{i}=\langle 1,0\rangle [/latex] and is directed along the positive horizontal axis. The vertical unit vector is written as [latex]\boldsymbol{j}=\langle 0,1\rangle [/latex] and is directed along the positive vertical axis.
Figure 14
A General Note: The Unit Vectors
If [latex]\boldsymbol{v}[/latex] is a nonzero vector, then [latex]\dfrac{\boldsymbol{v}}{|\boldsymbol{v}|}[/latex] is a unit vector in the direction of [latex]\boldsymbol{v}[/latex]. Any vector divided by its magnitude is a unit vector. Notice that magnitude is always a scalar, and dividing by a scalar is the same as multiplying by the reciprocal of the scalar.
Example: Finding the Unit Vector in the Direction of v
Find a unit vector in the same direction as [latex]\boldsymbol{v}=\langle -5,12\rangle [/latex].
