12.3-sm.dvi - 328 C H A P T E R 13 VECTOR GEOMETRY (ET...

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Unformatted text preview: 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 = a b a b , also cos < 0. By definition , but since cos < 0 then is in [ / 2 , ] . 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 = + = 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 be the angle between u and v . Which of the following is equal to the cos ?...
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This note was uploaded on 07/15/2009 for the course MATH 210 taught by Professor Hubscher during the Spring '08 term at University of Illinois at Urbana–Champaign.

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12.3-sm.dvi - 328 C H A P T E R 13 VECTOR GEOMETRY (ET...

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