# Further Applications of Trigonometry

## Vectors

An airplane is flying at an airspeed of 200 miles per hour headed on a SE bearing of 140°. A north wind (from north to south) is blowing at 16.2 miles per hour, as shown in . What are the ground speed and actual bearing of the plane?
Ground speed refers to the speed of a plane relative to the ground. Airspeed refers to the speed a plane can travel relative to its surrounding air mass. These two quantities are not the same because of the effect of wind. In an earlier section, we used triangles to solve a similar problem involving the movement of boats. Later in this section, we will find the airplane’s groundspeed and bearing, while investigating another approach to problems of this type. First, however, let’s examine the basics of vectors.

### A Geometric View of Vectors 

A vector is a specific quantity drawn as a line segment with an arrowhead at one end. It has an initial point, where it begins, and a terminal point, where it ends. A vector is defined by its magnitude, or the length of the line, and its direction, indicated by an arrowhead at the terminal point. Thus, a vector is a directed line segment. There are various symbols that distinguish vectors from other quantities:
1. Lower case, boldfaced type, with or without an arrow on top such as  

2. Given initial point 
  and terminal point 
  a vector can be represented as 
  The arrowhead on top is what indicates that it is not just a line, but a directed line segment.
3. Given an initial point of 
  and terminal point 
  a vector may be represented as 
  
This last symbol 
   has special significance. It is called the standard position. The position vector has an initial point 
  and a terminal point 
  To change any vector into the position vector, we think about the change in the x-coordinates and the change in the y-coordinates. Thus, if the initial point of a vector 
  is 
  and the terminal point is 
  then the position vector is found by calculating
In , we see the original vector 
  and the position vector 
 


### Finding Magnitude and Direction 

To work with a vector, we need to be able to find its magnitude and its direction. We find its magnitude using the Pythagorean Theorem or the distance formula, and we find its direction using the inverse tangent function.

### Performing Vector Addition and Scalar Multiplication

Now that we understand the properties of vectors, we can perform operations involving them. While it is convenient to think of the vector   
  as an arrow or directed line segment from the origin to the point 
  vectors can be situated anywhere in the plane. The sum of two vectors  and , or vector addition, produces a third vector + , the resultant vector.
To find  + , we first draw the vector , and from the terminal end of , we drawn the vector . In other words, we have the initial point of  meet the terminal end of . This position corresponds to the notion that we move along the first vector and then, from its terminal point, we move along the second vector. The sum  +  is the resultant vector because it results from addition or subtraction of two vectors. The resultant vector travels directly from the beginning of  to the end of  in a straight path, as shown in .
Vector subtraction is similar to vector addition. To find  − , view it as  + (−). Adding − is reversing direction of  and adding it to the end of  . The new vector begins at the start of   and stops at the end point of −. See  for a visual that compares vector addition and vector subtraction using parallelograms.

### Multiplying By a Scalar

While adding and subtracting vectors gives us a new vector with a different magnitude and direction, the process of multiplying a vector by a scalar, a constant, changes only the magnitude of the vector or the length of the line. Scalar multiplication has no effect on the direction unless the scalar is negative, in which case the direction of the resulting vector is opposite the direction of the original vector.

### Finding Component Form

In some applications involving vectors, it is helpful for us to be able to break a vector down into its components. Vectors are comprised of two components: the horizontal component is the 
  direction, and the vertical component is the 
  direction. For example, we can see in the graph in  that the position vector 
  comes from adding the vectors 1 and 2. We have 1 with initial point 
  and terminal point 
  
We also have 2 with initial point 
  and terminal point 
  
Therefore, the position vector is
Using the Pythagorean Theorem, the magnitude of 1 is 2, and the magnitude of 2 is 3. To find the magnitude of , use the formula with the position vector.
The magnitude of  is 
  To find the direction, we use the tangent function 
 

Thus, the magnitude of is 
  and the direction is 
  off the horizontal.

### Finding the Unit Vector in the Direction of v

In addition to finding a vector’s components, it is also useful in solving problems to find a vector in the same direction as the given vector, but of magnitude 1. We call a vector with a magnitude of 1 a unit vector. We can then 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   
  and is directed along the positive horizontal axis. The vertical unit vector is written as  
  and is directed along the positive vertical axis. See .

### Performing Operations with Vectors in Terms of i and j 

So far, we have investigated the basics of vectors: magnitude and direction, vector addition and subtraction, scalar multiplication, the components of vectors, and the representation of vectors geometrically. Now that we are familiar with the general strategies used in working with vectors, we will represent vectors in rectangular coordinates in terms of  and .


### Performing Operations on Vectors in Terms of i and j

When vectors are written in terms of and we can carry out addition, subtraction, and scalar multiplication by performing operations on corresponding components.

### Calculating the Component Form of a Vector: Direction 

We have seen how to draw vectors according to their initial and terminal points and how to find the position vector. We have also examined notation for vectors drawn specifically in the Cartesian coordinate plane using 
  For any of these vectors, we can calculate the magnitude. Now, we want to combine the key points, and look further at the ideas of magnitude and direction.
Calculating direction follows the same straightforward process we used for polar coordinates. We find the direction of the vector by finding the angle to the horizontal. We do this by using the basic trigonometric identities, but with replacing 


### Finding the Dot Product of Two Vectors 

As we discussed earlier in the section, scalar multiplication involves multiplying a vector by a scalar, and the result is a vector. As we have seen, multiplying a vector by a number is called scalar multiplication. If we multiply a vector by a vector, there are two possibilities: the dot product and the cross product. We will only examine the dot product here; you may encounter the cross product in more advanced mathematics courses.
The dot product of two vectors involves multiplying two vectors together, and the result is a scalar.

### Key Concepts

1. The position vector has its initial point at the origin. See .
2. If the position vector is the same for two vectors, they are equal. See .
3. Vectors are defined by their magnitude and direction. See .
4. If two vectors have the same magnitude and direction, they are equal. See .
5. Vector addition and subtraction result in a new vector found by adding or subtracting corresponding elements. See .
6. Scalar multiplication is multiplying a vector by a constant. Only the magnitude changes; the direction stays the same. See  and .
7. Vectors are comprised of two components: the horizontal component along the positive x-axis, and the vertical component along the positive y-axis. See .
8. The unit vector in the same direction of any nonzero vector is found by dividing the vector by its magnitude.
9. The magnitude of a vector in the rectangular coordinate system is 
  See .
10. In the rectangular coordinate system, unit vectors may be represented in terms of   and   where represents the horizontal component and represents the vertical component. Then,  = a + b  is a scalar multiple of by real numbers 
  See  and .
11. Adding and subtracting vectors in terms of i and j consists of adding or subtracting corresponding coefficients of i and corresponding coefficients of j. See .
12.  A vector v = a + b is written in terms of magnitude and direction as 
  See .
13. The dot product of two vectors is the product of the terms plus the product of the terms. See .
14. We can use the dot product to find the angle between two vectors.  and .
15. Dot products are useful for many types of physics applications. See .

### Section Exercises


### Verbal


### Algebraic

For the following exercises, determine whether the two vectors and are equal, where has an initial point 
  and a terminal point 
  and   has an initial point 
  and a terminal point 
 .
For the following exercises, use the vectors  =  + 5,  = −2− 3,  and  = 4 − .
For the following exercises, use the given vectors to compute  + ,  − , and 2 − 3.
For the following exercises, find a unit vector in the same direction as the given vector.
For the following exercises, find the magnitude and direction of the vector, 
 


### Graphical

For the following exercises, given draw 3 and 
  
For the following exercises, use the vectors shown to sketch  + ,  − , and 2.

For the following exercises, use the vectors shown to sketch 2 + .
For the following exercises, use the vectors shown to sketch  − 3.
For the following exercises, write the vector shown in component form.

### Extensions

For the following exercises, use the given magnitude and direction in standard position, write the vector in component form.

### Real-World Applications


### Chapter Review Exercises


### Non-right Triangles: Law of Sines

For the following exercises, assume 
  is opposite side 
  is opposite side 
  and 
  is opposite side 
  Solve each triangle, if possible. Round each answer to the nearest tenth.

### Non-right Triangles: Law of Cosines


### Polar Coordinates

For the following exercises, convert the given Cartesian equation to a polar equation.
For the following exercises, convert the given polar equation to a Cartesian equation.
For the following exercises, convert to rectangular form and graph.

### Polar Coordinates: Graphs

For the following exercises, test each equation for symmetry.

### Polar Form of Complex Numbers

For the following exercises, find the absolute value of each complex number.
Write the complex number in polar form.
For the following exercises, convert the complex number from polar to rectangular form.
For the following exercises, find the product 
  in polar form.
For the following exercises, find the quotient 
  in polar form.
For the following exercises, find the powers of each complex number in polar form.
For the following exercises, evaluate each root.
For the following exercises, plot the complex number in the complex plane.

### Parametric Equations

For the following exercises, eliminate the parameter 
  to rewrite the parametric equation as a Cartesian equation.

### Parametric Equations: Graphs

For the following exercises, make a table of values for each set of parametric equations, graph the equations, and include an orientation; then write the Cartesian equation.

### Vectors

For the following exercises, determine whether the two vectors, and are equal, where has an initial point 
  and a terminal point 
  and has an initial point 
  and a terminal point 
 

For the following exercises, use the vectors and to evaluate the expression.
For the following exercises, find a unit vector in the same direction as the given vector.
For the following exercises, find the magnitude and direction of the vector.
For the following exercises, calculate 


### Practice Test

Given 
  and 
  evaluate each expression.
For the following exercises, use the vectors  =  − 3 and  = 2 + 3.