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Chapter Two

Vectors

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Vectors

A quantity consisting only of magnitude is called a scalar quantity.

A quantity that has both magnitude and direction and obeys certain algebraic laws is called a vector quantity.

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Vector Components

See Figure 2-1.

or

See Figure 2-2. East: North:

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See Figure 2-3.

See Figures 2-4, 2-5, 2-6, 2-7.

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Example 2-1

A box is pulled by two persons exerting the forces F1 and F2 shown in Fig. 2-8, which F1 is given as 50 lb. Two questions may now be asked.

1. What force F2 must be applied so that the box moves only in the x direction?

2. What single force could replace F1 and F2 so that the box moves only in the x direction?

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Sol See Figure 2-9. Forces on x direction:

Forces on y direction:

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If the object is going to move in the x direction, then

Question 2 can be answered as

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Unit Vectors

See Figure 2-10. See Figure 2-11.

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Dot Product The dot product is defined as where on the right side A and B are the magnitud

e of each of the vectors and θ is the angle between them. See Figure 2-12.

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Cross Product

If C is the cross product of A and B, we have

and See Figure 2-13. The direction of C is perpendicular to both A and B and consequently perpendicular to the plane containing A and B (and obey the right-hand rule).

See Figures 2-14, 2-15. Homework: 12, 16, 17.

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