Vectors and Projectiles - Mission VP4 Detailed Help

An ant on a picnic table travels 18 cm eastward, then 20 cm northward and finally 12 cm westward. What is the magnitude of the ant's displacement (in cm)? 
 
(Note: Numbers are randomized numbers and likely different from the numbers listed here.)


 
An effective strategy for all questions in this mission will center around a rough sketch of the addition of two vectors (See Think About It section). Consider the following steps:
  • Begin by simplifying the east and the west displacements by adding these two vectors together. Consider west to be the negative direction and add the negative westward displacement to the positive eastward displacement. Record the result and label as ∑E-W (for sum of the east and west vectors).
  • Consider the sum of the east-west vectors (∑E-W) as a single vector and sketch it in the appropriate direction. Place an arrowhead at the end of the vector.
  • Starting at the arrowhead of the first vector, draw this second vector (the northward vector) in the appropriate direction and to the approximate length. Put an arrowhead at the end of the vector.
  • Draw the resultant vector from the tail of the first to the arrowhead of the second vector. Label the vector as R (for resultant) and put an arrowhead at the end of the resultant vector.
  • Since the resultant is a hypotenuse of a right triangle, the Pythagorean theorem can be used to calculate its magnitude (see Math Magic section).


 
The head-to-tail method of vector addition should be used to create a rough sketch of this physical situation. In this method, the first vector (the ∑E-W vectors) is sketched (not to scale) in its indicated direction. The second vector (20 cm, north) is then sketched (not to scale) starting at the head (arrowhead) of the first vector. The resultant vector is then drawn from the tail of the first vector to the head of the last vector. The Pythagorean Theorem can then be used to calculate the magnitude of the resultant (see Math Magic section).


 
Pythagorean Theorem
When two vectors that make a right angle to each other are added together, the resultant vector is the hypotenuse of a right triangle. The Pythagorean theorem can be used to calculate the magnitude of the resultant. If the right triangle has sides with lengths of x and y, then the length of the hypotenuse is the square root of the sum of the squares of the sides. That is,
 
hypotenuse = SQRT (x2+ y2)