5.19: Unit Vectors and Components
While working in your math class at school, the instructor passes everyone a map of Yourtown. She asks you to find your house and place a red dot on it, and then find the school and place a blue dot there. Your map looks like this:
She then asks you to break down the trip to your school in terms of component vectors and unit vectors. Are you able to do this?
Keep reading, and by the end of this Concept, you'll understand how to break a vector into its components and graph it using unit vector notation.
Watch This
James Sousa: Unit Vector Notation
Guidance
A unit vector is a vector that has a magnitude of one unit and can have any direction. Traditionally
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
A vector from the origin (0, 0) to the point (8, 0) is written as
The reason for having the component vectors perpendicular to one another is that this condition allows us to use the Pythagorean Theorem 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 or direction in space such as N, E, S or W is part of the problem.
Example A
What are the component vectors of the vector shown here?
Solution: Since the length of the vector is 5, and the angle the vector makes with the
And the "y" component is:
And we have the familiar 3, 4, 5 triangle, where the vector is the hypotenuse.
Example B
Why are unit vectors required when dealing with vector addition?
Solution:
Unit vectors are required because it is necessary to have like quantities for addition. If there are two numbers, they can be added. If there are two vectors, they can be added. But if you have a number and a vector, they can't be added. Having unit vectors along with a magnitude makes a quantity a vector.
Example C
What are the unit vectors and the lengths of the component vectors when
Solution:
The unit vectors in this case are
The length of the component vector in the
Vocabulary
Component Vectors: The component vectors of a given vector are two or more vectors whose sum is the given vector.
Unit Vector: A unit vector is a vector having a length of one.
Guided Practice
1. An inclined ramp is 12 feet long and forms an angle of
2. A wind vector has a magnitude of 25 miles per hour with an angle of
3. A vector
Solutions:
1.
2. Since the vector has an angle of
3. The "x" component is
Concept Problem Solution
In this Concept you learned that breaking a vector down into its components involves adding the portion of the vector along the "y" axis to the portion of the vector along the "x" axis. To accomplish this in the case of the map, you only need to write down the length the vector has in the "x" direction (along with an "x" unit vector) and then add to it the length the vector has in the "y" direction (along with a "y" unit vector). Your map should look like this:
Practice
 Describe how to find the vertical and horizontal components of a vector when given the magnitude and direction of the vector.

a⃗ has a magnitude of 6 and a direction of100∘ . Find the components of the vector. 
b⃗ has a magnitude of 3 and a direction of60∘ . Find the components of the vector. 
c⃗ has a magnitude of 2 and a direction of84∘ . Find the components of the vector. 
d⃗ has a magnitude of 5 and a direction of32∘ . Find the components of the vector.  \begin{align*}\vec{e}\end{align*} has a magnitude of 2 and a direction of \begin{align*}45^\circ\end{align*}. Find the components of the vector.
 \begin{align*}\vec{f}\end{align*} has a magnitude of 7 and a direction of \begin{align*}70^\circ\end{align*}. Find the components of the vector.
 A plane is flying on a bearing of \begin{align*}50^\circ\end{align*} at 450 mph. Find the component form of the velocity of the plane. What does the component form tell you?
 A baseball is thrown at a \begin{align*}20^\circ\end{align*} angle with the horizontal with an initial speed of 30 mph. Find the component form of the initial velocity.
 A plane is flying on a bearing of \begin{align*}300^\circ\end{align*} at 500 mph. Find the component form of the velocity of the plane.
 A plane is flying on a bearing of \begin{align*}150^\circ\end{align*} at 470 mph. At the same time, there is a wind blowing at a bearing of \begin{align*}200^\circ\end{align*} at 60 mph. What is the component form of the velocity of the plane?
 Using the information from the previous problem, find the actual ground speed of the plane.
 Wind is blowing at a magnitude of 50 mph with an angle of \begin{align*}25^\circ\end{align*} with respect to the east. What is the velocity of the wind blowing to the north? What is the velocity of the wind blowing to the east?
 Find a unit vector in the direction of \begin{align*}\vec{a}\end{align*}, a vector in standard position with terminal point (4, 3).
 Find a unit vector in the direction of \begin{align*}\vec{b}\end{align*}, a vector in standard position with terminal point (5, 1).
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Here you'll learn how to break down a vector into component vectors and unit vectors.