OBJECTIVES
1.  Students will be able to review and understand the principals of vector addition problems concerning vectors that are other than perpendicular.
2.  Students will solve for the magnitude of the resultant of vectors that are not perpendicular.
 

APPLICATIONS
In real life, Vectors can be applied in that they give you the ability to determine the magnitude and direction of a resultant of the displacement of an object.  For example, if a vessel is traveling in a direction with a specific velocity, and then changes that direction by an acute angle, it is possible to determine the resultant of these vectors, giving you the total displacement of the vessel.
 

CONCEPTS
 

• The line connecting two vectors in called the resultant.
• In order to solve for the resultant, form a right triangle with the origin and vector triangle and use the law of sines to solve for the dotted lines (see fig 1.).

• After forming the right triangle and finding the sides, use the Pythagorean Theorem to solve for the resultant.

FORMULAS

• Law of Sines

• Pythagorean Theorem

EXAMPLES

1.  An airplane is traveling west at 725 m/s.  The pilot then pitches the plane up 26 degrees north of west, into a climb.  Solve for the resulting displacement of the plane.

Create right triangle with x-axis.  See Figure 1.

Solve for the dotted lines through the Law of Sines.

655     =    y              655     =     x
sin90       sin64          sin90          sin26

Solve for the resultant through the Pythagorean Theorem.

                2                              2            2
287.13  +  1313.71  =  C              C  =  1344.723 m/s
 

2.  A hiker is traveling down a trail due east at 4 m/s.  He spots a bear to his right and changes his path 58 degrees north of east at a new pace of 17 m/s.  Solve for the resulting velocity of his two paths.

Create right triangle with x-axis.  See Figure 2.

Solve for the dotted lines through the Law of Sines.

17     =     y                 17     =     x
sin90       sin58            sin90       sin32

Solve for the resultant through the Pythagorean Theorem.

                  2                           2            2
14.417  +  13.009  =  C              C  =  19.419 m/s

3.  A ship is traveling west at a speed of 23 m/s.  The captain sees a storm on the horizon and changes the course 37 degrees north of west at an increased speed of 31 m/s to avoid the storm's path.  Solve for the displacement of the course of the ship.

Create right triangle with x-axis.  See Figure 3.

Solve for the dotted lines through the Law of Sines.

31    =     y                 31    =     x
sin90      sin37            sin90      sin53

Solve for the resultant through the Pythagorean Theorem.

               2                        2             2
24.76  +  18.66  =  C              C  =  31.002 m/s
 

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