CK-12 Trigonometry
Page 24
Assume is in standard position. For each terminal point is given, find the magnitude and direction of each vector.
Given the initial and terminal coordinates of , find the magnitude and direction. initial terminal
initial terminal
initial terminal
initial terminal
The magnitudes of vectors and are given along with the angle between them theta. Find the magnitude of the resultant and the angle it makes with .
Car is traveling at a speed of in a direction of . Car is traveling at a speed of in a direction of . If the two cars collide, what is the magnitude and direction of their resultant?
Two bulldozers are moving a boulder. One is pushing the boulder with of force and the other is pushing with of force. If the angle between the two forces is , what is the magnitude of the resultant and the direction made with the smaller force?
Review Answers
This is only true if both and are positive. If either or is negative, this will not be true.
The plane’s speed is and its direction is .
The magnitude is Newtons and the direction is .
The magnitude is and the direction is .
The magnitude is and the direction is .
Supplemental Links
http://hyperphysics.phy-astr.gsu.edu/hbase/vect.html
Vocabulary
directed line segment
A line segment having both magnitude and direction, often used to represent a vector.
displacement
When an object moves a certain distance in a certain direction.
equal vectors
Vectors with the same magnitude and direction.
force
When an object is pushed or pulled in a certain direction.
initial point
The starting point of a vector
magnitude
Length of a vector.
negative vector
A vector with the same magnitude as the original vector but with the opposite direction.
resultant
The sum of two or more vectors
standard position
A vector with its initial point at the origin of a coordinate plane.
terminal point
The ending point of a vector.
vector
Any quantity having magnitude and direction, often represented by an arrow.
velocity
When an object travels at a certain speed in a certain direction.
Component Vectors
Learning Objectives
A student will be able to:
Perform scalar multiplication with vectors.
Understand component vectors.
Find the resultant as a sum of two components.
Find the resultant as magnitude and direction.
Use component vectors to solve real-world and applied problems.
Introduction
Real-World Application:
A car has traveled in a direction of north of east. How far east of its initial point has it traveled? How far north has the car traveled?
We will refer back to this application later on.
The car traveled on a vector distance called a displacement. It moved in line at fixed distance from the starting point. Having two components in their expression, vectors are confusing to some. A diagram helps sort out confusions. Looking at vectors by separating them into components allows us to neatly a great many real-world problems. The components often relate to very different elements of the problem, such as wind speed in one direction and speed supplied by a motor in another.
In this section, we will learn about component vectors and how to find them. We will also explore other ways of finding the magnitude and direction of a resultant of two or more vectors. We will be using many of the tools we learned in the previous sections dealing with right and oblique triangles.
Vector Times a Scalar
In working with vectors there are two kinds of quantities employed. The first is the vector, a quantity that has both magnitude and direction. The second quantity is a scalar. Scalars are just numbers. The magnitude of a vector is a scalar quantity. A vector can be multiplied by a real number. This real number is called a scalar. The product of a vector and a scalar is a vector, written . It has the same direction as with a magnitude of if . If , the vector has the opposite direction of and a magnitude of .
Example 1: The speed of the wind before a hurricane arrived was from the SSE ( on the compass). It quadrupled when the hurricane arrived. What is the current vector for wind velocity? The wind is coming now at from the same direction.
Example 2: A sailboat was traveling at due north. After realize he had overshot his destination, the captain turned the boat around and began traveling twice as fast due south. What is the current vector for the speed of the ship? The ship is traveling at in the opposite direction.
If the vector is expressed in coordinates with the tip of the vector at origin, standard form, to scalar multiplication, we multiply our scalar by both the coordinates of our vector. The word scalar comes from “scale.” Seen from the origin, multiplying by a scalar just makes the vectors larger or smaller proportionally.
Example 3:
Consider the vector from the origin to . What would the representation of a vector that had three times the magnitude be? Here and = the directed segment from to .
The new coordinates of the directed segment are
What would happen if we had a negative value for ? How would this affect our vector?
Example 4: Consider the vector from the origin to . What would the representation of a vector that had the magnitude be?
Here, and the directed segment from to .
Since , our result would be a directed segment that is twice and long but in the opposite direction of our original vector.
Translation of Vectors and Slope
What would happen if we performed scalar multiplication on a vector that didn’t start at the origin?
Example 5: Consider the vector from to . What would the representation of a vector that had the magnitude be?
Here, and the directed segment from to .
Mathematically, two vectors are equal if their direction and magnitude are the same. The positions of the vectors do not matter. This means that if we have a vector that is not in standard position, we can translate it to the origin.
The initial point of is . In order to translate this to the origin, we would need to add to both the initial and terminal points of the vector.
Initial point:
Terminal point:
Now, to calculate :
The new coordinates of the directed segment are and . To translate this back to our original terminal point:
Initial point:
Terminal point:
The new coordinates of the directed segment are and .
Vectors with the same magnitude and direction are equal. This means that the same ordered pair could represent many different vectors. For instance, the ordered pair can represent a vector in standard position where the initial point is at the origin and the terminal point is at . This vector could be thought of as the resultant of a horizontal vector with a magnitude or and a vertical vector with a magnitude of Therefore, any vector with a horizontal component of and vertical component of could also be represented by the ordered pair .
All of these vectors have a horizontal component of and a vertical component of , even though they are in different positions on the coordinate plane.
If you think back to Algebra, you know that the slope of a line is the change in over the change in , or the vertical change over the horizontal change. Looking at our vectors above, since they all have the same horizontal and vertical components, they all have the same slope, even though they do not all start at the origin.
Unit Vectors and Components
A unit vector is a vector that has a magnitude of one unit and can have any direction. Traditionally is the unit v
ector in the direction and is the unit vector in the direction. and . Unit vectors on perpendicular axes can be used to express all vectors in that plane.” Vectors are used to express position and motion in three dimensions with as the unit vector in the direction. We are not studying space in this course. The unit vector notation may seem burdensome but one must distinguish between a vector and the components of that vector in the direction of the or axis. The unit vectors carry the meaning for the direction of the vector in each of the coordinate directions. The number in front of the unit vector shows its magnitude or length. Unit vectors are convenient if one wishes to express a or vector as a sum of two or three orthogonal components, such as and axes, or the axis.
Component vectors of a given vector are two or more vectors whose sum is the given vector. The sum is viewed as equivalent to the original vector. Since component vectors can have any direction, it is useful to have them perpendicular to one another. Commonly one chooses the and axis as the basis for the unit vectors. Component vectors do not have to be orthogonal.
A vector from the origin to the point is written as . A vector from the origin to the point is written as .
The reason for having the component vectors perpendicular to one another is that this condition allows us to use the Theorem of Pythagoras and trigonometric ratios to find the magnitude and direction of the components. One can solve vector problems without use of unit vectors if specific information about orientation, direction in space such as or are not part of the problem.
Resultant as the Sum of Two Components
We can look at any vector as the resultant of two perpendicular components. In the figure below, is the horizontal component of and is the vertical component of . Therefore must be some magnitude times the unit vector in the direction.
The sum of vector plus vector is: . This addition can also be written as .
From this figure, we can see how would be the resultant if we added and together using the triangle method. If we are given the vector , we can find the components of , , and using trigonometric rations if we know the magnitude and direction of .
Example 6: (refer to the figure above)
If and its direction is find the horizontal and vertical components.
If we know an angle and a side of a right triangle, we can find the other remaining sides using trigonometric ratios. In this case, is the hypotenuse of our triangle, is the side adjacent to our angle, is the side opposite our angle, and is directed along the axis.
To find , we will use cosine since we are using the adjacent side and the hypotenuse. Please note this is a scalar equation so all quantities are just numbers. It is written as the quotient of the magnitudes, not the vectors.
To find , we will use sine since we are using the opposite side and the hypotenuse.
Answer: The horizontal component is and the vertical component is . One can rewrite this in vector notation as
We will now refer back to the application at the beginning of the section.
A car has traveled in a direction of north of east. How far east of its initial point has it traveled? How far north has he traveled?
In order to find how far the car has traveled east and how far it has traveled north, we will need to find the horizontal and vertical components of the vector.
To find :
To find :
Answer: The car has traveled east and north of its original destination. In a vector equation it is displacement.
Resultant as Magnitude and Direction
If we don’t have two perpendicular vectors, we can still find the magnitude and direction of the resultant without a graphic estimate with a construction using a compass and ruler. This can be accomplished using both the Law of Sines and the Law of Cosines.
Example 7:
Vector makes a angle with vector . The magnitude of is . The magnitude of is . Find the magnitude and direction the resultant makes with the smaller vector.
There is no preferred orientation such as a compass direction or any necessary use of and coordinates. The problem can be solved without use of unit vectors
In order to solve this problem, we will need to use the parallelogram method. Since vectors only have magnitude and direction, one can move them on the plane to any position one wishes, as long as the magnitude and direction remain the same First, we will complete the parallelogram: Label the vectors. Move vector so its tail is on the tip of vector . Move vector so its tail is on the tip of vector . This makes a parallelogram because the angles did not change during the translation. Put in labels for the vertices of the parallelogram.
Since opposite angles in a parallelogram are congruent, we know that opposite angles in a parallelogram are congruent, we can find angle .
Now, we know two sides and the included angle in an oblique triangle. This means we can use the Law of Cosines to find the magnitude of our resultant.
To find the direction, we can use the Law of Sines since we now know an angle and a side across from it. We choose the Law of Sines because it is a proportion and less computationally intense than the Law of Cosines.
Answer: The magnitude of the resultant is and the direction it makes with the smaller force is counterclockwise.
We can use a similar method to add three or more vectors.
Example 8: Vector makes a angle with the horizontal and has a magnitude of . Vector makes a angle with the horizontal and has a magnitude of . Vector makes a angle with the horizontal and has a magnitude of . Find the magnitude and direction (with the horizontal) of the resultant of all three vectors.
To begin this problem, we will find the resultant using Vector and Vector . We will do this using the parallelogram method like we did above.
Since Vector makes a angle with the horizontal and Vector makes a angle with the horizontal, we know that the angle between the two is .
To find :
Now, we will use the Law of Cosines to find the magnitude of .
Next, we will use the Law of Sines to find the measure of angle .
We know that Vector forms a angle with the horizontal so we add that value to the measure of to find the angle makes with the horizontal. Therefore, makes a angle with the horizontal.
Next, we will take , and we will find the resultant vector of and Vector from above. We will repeat the same process we used above.
Vector makes a angle with the horizontal and makes a angle with the horizontal. This means that the angle between the two is . We will use this information to find the measure of .
Now we will use the Law of Cosines to find the magnitude of .
Next, we will use the Law of Sines to find .
Finally, we will take the measure of and add it to the angle that forms with the horizontal. Therefore, forms a angle with the horizontal.
Answer: The resultant has a magnitude of and forms a angle with the horizontal.
Applications
Real-World Application: Two forces of and are acting on an object. The angle between the two forces is . What is the magnitude of the resultant? What angle does the resultant make with the smaller force?
We do not need unit vectors here as there is no preferred direction like a compass direction or a specific axis. First, to find the magnitude we will need to figure out the other angle in our parallelogram.
Now that we know the other angle, we can find the magnitude using the Law of Cosines.
To find the angle the resultant makes with the smaller force, we will use the Law of Sines.
Answer: The magnitude of the resultant is and the resultant makes an angle of counterclockwise with the smaller force.
Application:
Two trucks are pulling a large chunk of stone. Truck is pulling with a force of at a angle from the horizontal while Truck is pulling with a force of at a angle from the horizontal. What is the magnitude and direction of the resultant force?
Since Truck has a direction of and Truck has a direction of , we can see that the angle between the two forces is . W
e need this angle measurement in order to figure out the other angles in our parallelogram.
Now, use the Law of Cosines to find the magnitude of the resultant.
Now to find the direction we will use the Law of Sines.
Since we want the direction we need to add the to the of the smaller force.
Answer: The magnitude is and clockwise from the horizontal.
Points to Consider
How you can verify if your answers to problems involving vectors that are not perpendicular are correct?
In what ways are solving problems with oblique triangles and solving problems involving vectors similar?
In what ways are the different?
When is it appropriate to use vectors instead of oblique triangles to solve problems?
When is it helpful to use unit vectors? When can one solve without explicitly using them?