

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 ComponentsA vector is a quantity which has both magnitude and direction. Displacement, velocity, acceleration, and force are the vector quantities which we have discussed thus far in the Physics Classroom Tutorial. In the first couple of units, all vectors which we discussed were simply directed up, down, left or right. When there was a freebody diagram depicting the forces acting upon an object, each individual force was directed in one dimension  either up or down or left or right. When an object had an acceleration and we described its direction, it was directed in one dimension  either up or down or left or right. Now in this unit, we begin to see examples of vectors which are directed in two dimensions  upward and rightward, northward and westward, eastward and southward, etc. In situations in which vectors are directed at angles to the customary coordinate axes, a useful mathematical trick will be employed to transform the vector into two parts with each part being directed along the coordinate axes. For example, a vector which is directed northwest can be thought of as having two parts  a northward part and a westward part. A vector which is directed upward and rightward can be thought of as having two parts  an upward part and a rightward part. Any vector directed in two dimensions can be thought of as having an influence in two different directions. That is, it can be thought of as having two parts. Each part of a twodimensional vector is known as a component. The components of a vector depict the influence of that vector in a given direction. The combined influence of the two components is equivalent to the influence of the single twodimensional vector. The single twodimensional vector could be replaced by the two components.
If Fido's dog chain is stretched upward and rightward and pulled tight by his master, then the tension force in the chain has two components  an upward component and a rightward component. To Fido, the influence of the chain on his body is equivalent to the influence of two chains on his body  one pulling upward and the other pulling rightward. If the single chain were replaced by two chains. with each chain having the magnitude and direction of the components, then Fido would not know the difference. This is not because Fido is dumb (a quick glance at his picture reveals that he is certainly not that), but rather because the combined influence of the two components is equivalent to the influence of the single twodimensional vector.
Consider a picture which is hung to a wall by means of two wires which are stretched vertically and horizontally. Each wire exerts a tension force upon the picture to support its weight. Since each wire is stretched in two dimensions (both vertically and horizontally), the tension force of each wire has two components  a vertical component and a horizontal component. Focusing on the wire on the left, we could say that the wire has a leftward and an upward component. This is to say that the wire on the left could be replaced by two wires, one pulling leftward and the other pulling upward. If the single wire were replaced by two wires (each one having the magnitude and direction of the components), then there would be no affect upon the stability of the picture. The combined influence of the two components is equivalent to the influence of the single twodimensional vector.
Consider an airplane which is flying from Chicago's O'Hare International Airport to a destination in Canada. Suppose that the plane is flying in such a manner that its resulting displacement vector is northwest. If this is the case, then the displacement of the plane has two components  a component in the northward direction and a component in the westward direction. This is to say that the plane would have the same displacement if it were to take the trip into Canada in two segments  one directed due North and the other directed due West. If the single displacement vector were replaced by these two individual displacement vectors, then the passengers in the plane would end up in the same final position. The combined influence of the two components is equivalent to the influence of the single twodimensional displacement. Any vector directed in two dimensions can be thought of as having two different components. The component of a single vector describes the influence of that vector in a given direction. In the next part of this lesson, we will investigate two methods for determining the magnitude of the components. That is, we will investigate how much influence a vector exerts in a given direction. 
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