Objectives
After completing this chapter, you should be able to:
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Understand the dot product as a way of combining two vectors into a single number that measures the extent to which they point in the same direction.
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Interpret the sign of the dot product geometrically—positive for an acute angle, negative for an obtuse angle, and zero for perpendicular vectors—and recognize orthogonality as the vanishing of the dot product.
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Understand how the dot product encodes the angle between two vectors, connecting the algebra of vectors to the Law of Cosines.
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Interpret the projection of one vector onto another as the component of one vector in the direction of the other, and understand the decomposition of a vector into parts parallel and perpendicular to a given direction.
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Understand work as a physical measure of the effect of a force along a direction of motion, and appreciate why the dot product is the natural tool for expressing it.
