

Lesson 1: Vectors  Fundamentals and OperationsRelative Velocity and Riverboat Problems
Independence of Perp. Components of
Motion Lesson 2: Projectile MotionCharacteristics of a Projectile's Trajectory Describing Projectiles with Numbers: Horizontal and Vertical Velocity Horiz. Launched Projectiles  Problem Solving
NonHoriz. Launched Projectiles 
Problem Solving Lesson 3 : Forces in Two Dimensions

Lesson 1: Vectors  Fundamentals and OperationsVector ResolutionAs mentioned earlier in
this lesson, any vector directed at an angle to the
horizontal (or the vertical) can be thought of as having two
parts (or components). That is, any vector directed in two
dimensions can be thought of as having two components. For
example, if a chain pulls upward at an angle on the collar
of a dog, then there is a tension force directed in two
dimensions. This tension force has two components: an upward
component and a rightward component. As another example,
consider an airplane which is displaced northwest from
O'Hare International Airport (in Chicago) to a destination
in Canada. The displacement vector of the plane is in two
dimensions (northwest). Thus, this displacement vector has
two components: a northward component and a westward
component. In this unit, we learn two basic methods for determining the magnitudes of the components of a vector directed in two dimensions. The process of determining the magnitude of a vector is known as vector resolution. The two methods of vector resolution which we will examine are
Parallelogram Method of Vector ResolutionThe parallelogram method of vector resolution involves using an accurately drawn, scaled vector diagram to determine the components of the vector. Briefly put, the method involves drawing the vector to scale in the indicated direction, sketching a parallelogram around the vector such that the vector is the diagonal of the parallelogram, and determining the magnitude of the components (the sides of the parallelogram) using the scale. If one desires to determine the components as directed along the traditional x and ycoordinate axes, then the parallelogram is a rectangle with sides which stretch vertically and horizontally. A stepbystep procedure for using the parallelogram method of vector resolution is:
The stepbystep procedure above is illustrated in the diagram below to show how a velocity vector with a magnitude of 50 m/s and a direction of 60 degrees above the horizontal may be resolved into two components. The diagram shows that the vector is first drawn to scale in the indicated direction; a parallelogram is sketched about the vector; the components are labeled on the diagram; and the result of measuring the length of the vector components and converting to m/s using the scale. (NOTE: because different computer monitors have different resolutions, the actual length of the vector on your monitor may not be 5 cm.)
Trigonometric Method of Vector ResolutionThe trigonometric method of vector resolution involves using trigonometric functions to determine the components of the vector. Earlier in lesson 1, the use of trigonometric functions to determine the direction of a vector was described. Now in this part of lesson 1, trigonometric functions will be used to determine the components of a single vector. Recall from the earlier discussion that trigonometric functions relate the ratio of the lengths of the sides of a right triangle to the measure of an acute angle within the right triangle. As such, trigonometric functions can be used to determine the length of the sides of a right triangle if an angle measure and the length of one side are known. The method of employing trigonometric functions to determine the components of a vector are as follows:
The above method is illustrated below for determining the components of the force acting upon Fido. As the 60Newton tension force acts upward and rightward on Fido at an angle of 40 degrees, the components of this force can be determined using trigonometric functions.
In conclusion, a vector directed in two dimensions has two components  that is, an influence in two separate directions. The amount of influence in a given direction can be determined using methods of vector resolution. Two methods of vector resolution have been described here  a graphical method (parallelogram method) and a trigonometric method.

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