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Unformatted text preview: LESSON 2 Vector Algebra READ: Sections 13.3, 13.4 NOTES: Besides adding, subtracting, and multiplying vectors by scalars, there are two other useful operations with vectors. The scalar product defined in the previous lesson combines a scalar and a vector to produce a new vector. The dot product combines two vectors to produce a scalar, and the cross product combines two vectors to produce another vector. To form the dot product of two vectors, multiply the corresponding components of the vectors, and add up the resulting numbers. In symbols, if-→ v = h a 1 ,a 2 ,a 3 i , and-→ w = h b 1 ,b 2 ,b 3 i , then-→ v ·-→ w = a 1 b 1 + a 2 b 2 + a 3 b 3 . As usual, when a new operation is introduced, the algebraic behavior of the operation ought to be spelled out. Is the operation commutative, for example. In other words, is it true that-→ v ·-→ w =-→ w ·-→ v ? This, as well as the other natural algebraic questions about dot products are listed in the table on page 676 of the text. The proofs are all very simple: just write out the left and right side of each equation, and check to see the two sides really are the same. For example, to show the operation is commutative (which is property 2 in the table), let’s suppose-→ v = h a 1 ,a 2 ,a 3 i , and-→ w = h b 1 ,b 2 ,b 3 i . Then-→ v ·-→ w = a 1 b 1 + a 2 b 2 + a 3 b 3 . On the other hand-→ w ·-→ v = b 1 a 1 + b 2 a 2 + b 3 a 3 . Comparing the two results, we see they are the same, and so-→ v ·-→ w =-→ w ·-→ v . The other proofs go the same way. Try them. Notice that properties (iii) and (iv) show how dot products interact with vector addition and scalar multiplication. One particularly useful formula in the list says that the magnitude of vector is the square root of the dot product of the vector with itself. In other words, ||-→ v || = √-→ v ·-→ v . It always helps the understanding to be able to visualize concepts. The dot product provides us with a tool to aid visualization. When two vectors are drawn with their initial points at the origin, it makes sense to ask about the angle between the vectors, at least if neither one is the zero vector-→ 0 = h , , i . The angle between two vectors is always taken to be a number between 0 and π . The dot product can be used to calculate that angle. Using the Law of Cosines, and checking the diagram on page 676 of the text, it’s not too hard to see that-→ v ·-→ w = ||-→ v ||||-→ w || cos θ , where θ is the angle between-→ v and-→ w . That equation provides a means of computing θ . It also helps us understand a bit about what the dot product means geometrically. Think about that equation and imagine the effect on the value of the dot product of-→ v and-→ w as we hold-→ v in a fixed position, and allow-→ w to rotate. When-→ v and-→ w point in the same direction, the angle between them will be 0, and so the value of cos θ will be 1, and the value of-→ v ·-→ w will be fairly large. Aswill be fairly large....
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