• Component Method of Summing Vectors

Pre-requisite skills:  Understanding of Pythagorean Theorem.  Understanding of the arctangent helpful.

Approximate completion time:  Under an hour.

Provide sufficient detail to verify that the assignment was completed in a meaningful manner.

Applet by Vladimir Sorokin

The purpose of this assignment is to establish three very important principles of vector addition: 

• Summing vectors that point in arbitrary directions is not easy.

• However, summing vectors when they are pointing along the same line is very easy. 

• And summing vectors that are perpendicular to each other can be performed using the simplest rules of geometry.

Vectors Pointing in Arbitrary Directions

1.  Using the mouse, draw a vector that is six units of length in the x-direction and eight units of length in the y-direction.  Call this vector A.

2.  Now draw a second vector B that is one unit of length in the x-direction and four units of length in the x-direction. 

Notice that vectors A and B are not parallel, nor are they perpendicular.

3. Choose the A+B button on the left-side of the applet.  The applet will draw a resultant vector in black.  

This applet uses the parallelogram method for displaying the resultant vector.

Let us see if we can find the lengths and directions of the resultant vector using simple geometry or trigonometry.

4.  Add the lengths of vectors A and B (use the exact values given on the right-side of the applet).  Does this length agree with the length of the resultant vector shown in the applet?

5.  Let us try the Pythagorean theorem.   Calculate the following:

where A and B are the lengths of vectors A and B. 

(a) Does this length agree with the length of the resultant vector in the applet?

(b) Why did our simple attempts to find the lengths of the resultant vector fail in both situations?

Parallel Vectors

Vectors pointing in the same direction

1. Using the mouse, draw a vector of reasonable length and oriented along some arbitrary direction.  Write down the components of this vector on a sheet of paper.

2.  Now draw a second vector that points along the same direction.  Again, write down the components.

3.  Choose the A+B button on the left-side of the applet.  The applet will draw a resultant vector in black.  

(a) In which direction does the resultant vector point?

(b) What is its length?

(c) How is its length related to the lengths of the two vectors, A and B?

Vectors pointing in the opposite direction.  (Sometimes called "anti-parallel vectors.")

1.  Using the mouse, draw a vector of reasonable length and oriented along some arbitrary direction.  Call this vector A.  Write down the components of A on a sheet of paper.

2.  Now draw a second vector B that points in the opposite direction, but shorter in length.  Again, write down the components of B.

3.  Choose the A+B button on the left-side of the applet.  The applet will draw a resultant vector in black.  

(a)  In which direction does the resultant vector point?

(b)  What is its length?

(c)  How is its length related to the lengths of the two vectors, A and B?

4. Repeat the above steps, but now make vector B longer than vector A.   Now in which direction does the resultant vector point?

5.  From what you see, write a brief sentence that describes the general rule of adding vectors that point along the same line (either in the same direction or in opposite directions).

Perpendicular Vectors

1. Using the mouse, draw a vector that is five units of length in the x-direction and three units of length in the y-direction.  Call this vector A.

2.  Now draw a second vector B that is three units of length in the x-direction and five units of length in the x-direction.  Vectors A and B should be perpendicular.

3.  Choose the A+B button on the left-side of the applet.  The applet will draw a resultant vector in black.  

4.  This applet uses the parallelogram method for displaying the resultant vector.   When the two vectors A and B are perpendicular, what shape does this parallelogram assume?

5.  Notice that the vectors A and B and the resultant vector form a triangle.  What special type of triangle appears?  

6. Using the Pythagorean Theorem, caclulate the length of the resultant vector.   (Show all work.)  Compare the length found in your calculation with the length of the resultant vector given in the applet.  Are they the same?   Why does the Pythagorean Theorem work in this case?  (Hint: Think in terms of right triangles.)

7.  From what you see, write a brief sentence that describes the general rule of adding vectors that point along the same line (either in the same direction or in opposite directions).

Helpful Resources

1. Vector Addition and Subtraction by Wesley Dean.
2. Vectors and Motion in Two Dimensions by Tom Henderson.
3. Vector Illuminations by Ron Greene.
4. Sum of Vectors by Phillip Dukes. (Assignment)
5. Vector Math Worksheet by Gary Richert (Assignment)
6. Vectors by Angel Franco Garcia. (in Spanish)
7. Zona Land's Vectors by Ed Zobel.

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