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12.3-sm.dvi - 328 C H A P T E R 13 V EC T OR G E OME TRY(ET...

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328 C H A P T E R 13 VECTOR GEOMETRY (ET CHAPTER 12) 13.3 Dot Product and the Angle Between Two Vectors (ET Section 12.3) Preliminary Questions 1. Is the dot product of two vectors a scalar or a vector? SOLUTION The dot product of two vectors is the sum of products of scalars, hence it is a scalar. 2. What can you say about the angle between a and b if a · b < 0? SOLUTION Since the cosine of the angle between a and b satisfies cos θ = a · b a b , also cos θ < 0. By definition 0 θ π , but since cos θ < 0 then θ is in [ π / 2 , π ] . In other words, the angle between a and b is obtuse. 3. Suppose that v is orthogonal to both u and w . Which property of dot products allows us to conclude that v is orthogonal to u + w ? SOLUTION One property is that two vectors are orthogonal if and only if the dot product of the two vectors is zero. The second property is the Distributive Law. Since v is orthogonal to u and w , we have v · u = 0 and v · w = 0. Therefore, v · ( u + w ) = v · u + v · w = 0 + 0 = 0 We conclude that v is orthogonal to u + w . 4. What is proj v ( v ) ? SOLUTION The projection of v along itself is v , since proj v ( v ) = v · v v · v v = v 5. What is the difference, if any, between the projection of u along v and the projection along the unit vector e v ? SOLUTION The projection of u along v is the vector proj v ( u ) = ( u · e v ) e v (1) The projection of u along e v is the vector proj e v ( u ) = ( u · e e v ) e e v (2) Since e e v = e v , the vectors in (1) and (2) are identical. That is, the projection of u along v is the projection of u along e v . 6. Suppose that proj v ( u ) = v . Determine: (a) proj 2 v ( u ) (b) proj v ( 2 u ) SOLUTION (a) The projection of u along 2 v is the following vector: proj 2 v ( u ) = u · 2 v ( 2 v ) · ( 2 v ) 2 v = 2 u · v 4 v · v 2 v = 4 u · v 4 v · v v = u · v v · v v = proj v ( u ) = v (b) The projection of 2 u along v is the following vector: proj v ( 2 u ) = 2 u · v v · v v = 2 u · v v · v v = 2proj v ( u ) = 2 · v = 2 v 7. Let u || be the projection of u along v . Which of the following is the projection u along the vector 2 v ? (a) 1 2 u || (b) u || (c) 2 u || SOLUTION Since u is the projection of u along v , we have, u = u · v v · v v The projection of u along the vector 2 v is proj 2 v ( u ) = u · 2 v 2 v · 2 v 2 v = 2 u · v 4 v · v 2 v = 4 u · v 4 v · v v = u · v v · v v = u That is, u is the projection of u along 2 v . Notice that the projection of u along v is the projection of u along the unit vector e v , hence it depends on the direction of v rather than on the length of v . Therefore, the projection of u along v and along 2 v is the same vector. 8. Let θ be the angle between u and v . Which of the following is equal to the cos θ ? (a) u · v (b) u · e v (c) e u · e v SOLUTION By the Theorems on the Dot Product and the Angle Between Vectors, we have cos θ = u · v u v = u u · v v = e u · e v The correct answer is (c).

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S E C T I O N 13.3 Dot Product and the Angle Between Two Vectors (ET Section 12.3) 329 Exercises In Exercises 1–12, compute the dot product.
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12.3-sm.dvi - 328 C H A P T E R 13 V EC T OR G E OME TRY(ET...

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