ss_8_5

ss_8_5 - Section 8.5 The Dot Product The dot product of two...

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Unformatted text preview: Section 8.5 The Dot Product The dot product of two vectors is a scalar (number) that is useful in several applications. In particular, the dot product provides an easy means for determining whether or not two vectors are parallel or perpendicular, and for finding the angle between two vectors. Let v = v 1 i + v 2 j and w = w 1 i + w 2 j be two vectors. The dot product, v · w , is defined as follows. • v · w = v 1 w 1 + v 2 w 2 Dot Product Example . If v = 2 i- 3 j and w =- i + 2 j , then the dot product v · w = (2)(- 1) + (- 3)(2) =- 2- 6 =- 8. Example . Let v = 2 i- 3 j , then v · v = (2)(2) + (- 3)(- 3) = 13 = || v || 2 . Note: v · v = || v || 2 not only holds in this example, it holds in general. Properties of the Dot Product • u · v = v · u • u · ( v + w ) = u · v + u · w • v · v = || v || 2 • v · = 0 Theorem. u · v = || u |||| v || cos θ , where θ is the angle between u and v . Hence • cos θ = u · v || u |||| v || Note: The angle θ between two vectors is the smaller of the two possible angles. Thus 0 ≤ θ ≤ π . Example . Find the angle between the two vectors u = 4 i- 3 j and v = 2 i +5 j , shown in the following diagram....
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This note was uploaded on 12/11/2011 for the course MAC 1140 taught by Professor Kutter during the Fall '11 term at FSU.

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ss_8_5 - Section 8.5 The Dot Product The dot product of two...

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