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Unit Vectors and Components

Vectors with length of one; two or more vectors whose sum is the given vector.

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Unit Vectors and Components

While working in your math class at school, the instructor passes everyone a map of Your town. 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?

Unit Vectors and Components

A unit vector is a vector that has a magnitude of one unit and can have any direction. Traditionally \begin{align*}\hat{i}\end{align*} (read “\begin{align*}i\end{align*} hat”) is the unit vector in the \begin{align*}x\end{align*} direction and \begin{align*}\hat{j}\end{align*} (read “\begin{align*}j\end{align*} hat”) is the unit vector in the \begin{align*}y\end{align*} direction. \begin{align*}|\hat{i}| = 1\end{align*} and \begin{align*}|\hat{j}| = 1\end{align*}. 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 \begin{align*}\hat{k}\end{align*} (“\begin{align*}k\end{align*} hat”) as the unit vector in the \begin{align*}z\end{align*} direction.

We are not studying 3D 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 \begin{align*}x-\end{align*} or \begin{align*}y-\end{align*}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 2D or 3D vector as a sum of two or three orthogonal components, such as \begin{align*}x-\end{align*} and \begin{align*}y-\end{align*}axes, or the \begin{align*}z-\end{align*}axis. (Orthogonal components are those that intersect at right angles.)

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 \begin{align*}x\end{align*} and \begin{align*}y\end{align*} axis as the basis for the unit vectors. Component vectors do not have to be orthogonal.

A vector from the origin (0, 0) to the point (8, 0) is written as \begin{align*}8 \hat{i}\end{align*}. A vector from the origin to the point (0, 6) is written as \begin{align*}6 \hat{j}\end{align*}.

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.

Let's take a look at a few example problems. 

1. What are the component vectors of the vector shown here?

Since the length of the vector is 5, and the angle the vector makes with the \begin{align*}x\end{align*} axis is \begin{align*}53.13^\circ\end{align*}, the "x" component of the vector is:

\begin{align*} |V_x| = |\vec{V}|\cos 53.13^\circ\\ |V_x| = (5)(.6) = 3\\ \end{align*}

And the "y" component is:

\begin{align*} |V_y| = |\vec{V}|\sin 53.13^\circ\\ |V_y| = (5)(.8) = 4\\ \end{align*}

And we have the familiar 3, 4, 5 triangle, where the vector is the hypotenuse.

2. Why are unit vectors required when dealing with vector addition?

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.

3. What are the unit vectors and the lengths of the component vectors when

\begin{align*}\vec{V} = 7 \hat{i} + 9 \hat{j}\end{align*}

The unit vectors in this case are \begin{align*}\hat{i}\end{align*} and \begin{align*}\hat{j}\end{align*}. In some courses and books you might see the notation for unit vectors written instead as \begin{align*}\hat{x}\end{align*} and \begin{align*}\hat{y}\end{align*}.

The length of the component vector in the \begin{align*}\hat{i}\end{align*} direction is 7, and the component vector in the \begin{align*}\hat{j}\end{align*} direction is 9.

Examples

Example 1

Earlier, you were asked to break down the trip to your school in terms of component vectors and unit vectors. 

In this section, 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:

Example 2

 An inclined ramp is 12 feet long and forms an angle of \begin{align*}28.2^\circ\end{align*} with the ground. Find the horizontal and vertical components of the ramp.

\begin{align*}y = 12 \sin 28.2^\circ = 5.7, x = 12 \cos 28.2^\circ = 10.6\end{align*}

Example 3

A wind vector has a magnitude of 25 miles per hour with an angle of \begin{align*}20^\circ\end{align*} with respect to the east. Determine how much the wind is blowing to the north and how much it is blowing to the east.

Since the vector has an angle of \begin{align*}20^\circ\end{align*} with respect to the horizontal, the component to the east is \begin{align*}25 \cos 20^\circ = 23.49\end{align*} miles per hour. In the same way, the component to the north is \begin{align*}25 \sin 20^\circ = 8.55\end{align*} miles per hour. 

Example 4

A vector \begin{align*}|\vec{V}|\end{align*} has a magnitude of 25 inches, and is at an angle of \begin{align*}80^\circ\end{align*} with respect to the positive "x" axis. Write the vector in component and unit vector notation.

The "x" component is \begin{align*}25 \cos 80^\circ = 4.34\end{align*}. The "y" component is \begin{align*}25 \sin 80^\circ = 24.62\end{align*}. Therefore, the vector can be written as \begin{align*}|\vec{V}| = 4.34 \hat{i} + 24.62 \hat{j}\end{align*}.

Review

  1. Describe how to find the vertical and horizontal components of a vector when given the magnitude and direction of the vector.
  2. \begin{align*}\vec{a}\end{align*} has a magnitude of 6 and a direction of \begin{align*}100^\circ\end{align*}. Find the components of the vector.
  3. \begin{align*}\vec{b}\end{align*} has a magnitude of 3 and a direction of \begin{align*}60^\circ\end{align*}. Find the components of the vector.
  4. \begin{align*}\vec{c}\end{align*} has a magnitude of 2 and a direction of \begin{align*}84^\circ\end{align*}. Find the components of the vector.
  5. \begin{align*}\vec{d}\end{align*} has a magnitude of 5 and a direction of \begin{align*}32^\circ\end{align*}. Find the components of the vector.
  6. \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.
  7. \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.
  8. 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?
  9. 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.
  10. 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.
  11. 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?
  12. Using the information from the previous problem, find the actual ground speed of the plane.
  13. 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?
  14. Find a unit vector in the direction of \begin{align*}\vec{a}\end{align*}, a vector in standard position with terminal point (-4, 3).
  15. Find a unit vector in the direction of \begin{align*}\vec{b}\end{align*}, a vector in standard position with terminal point (5, 1).

Review (Answers)

To see the Review answers, open this PDF file and look for section 5.19. 

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