# Assoc prof nguyen ngoc hai calculus ii vectors and

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Assoc. Prof. Nguyen Ngoc Hai CALCULUS II VECTORS AND THE GEOMETRY OF SPACE
2.2 VECTORS 2.2.3 THE DOT PRODUCT Example 2.5 Find the scalar projection and vector projection of b = h 1 , 1 , 2 i onto a = h- 2 , 3 , 1 i . Solution Since a = p ( - 2) 2 + 3 2 + 1 2 = 14, comp a b = a · b | a | = ( - 2) × 1 + 3 × 1 + 1 × 2 14 = 3 14 . Thus proj a b = 3 14 a | a | = 3 14 a = D - 3 7 , 9 14 , 3 14 E . Assoc. Prof. Nguyen Ngoc Hai CALCULUS II VECTORS AND THE GEOMETRY OF SPACE
2.2 VECTORS 2.2.3 THE DOT PRODUCT One use of projections occurs in physics in calculating work. If the force moves the object from P to Q , then the displacement vector is -→ PQ . Definition 2.4 The work done by a constant force acting through a displacement -→ PQ is Work = F · -→ PQ = | || -→ PQ | cos θ. Assoc. Prof. Nguyen Ngoc Hai CALCULUS II VECTORS AND THE GEOMETRY OF SPACE
2.2 VECTORS 2.2.3 THE DOT PRODUCT Example 2.6 A wagon is pulled a distance of 100 m along a horizontal path by a constant force of 70 N. The handle of the wagon is held at an angle of 35 above the horizontal. Find the work done by the force. FIGURE 2.22 Assoc. Prof. Nguyen Ngoc Hai CALCULUS II VECTORS AND THE GEOMETRY OF SPACE
2.2 VECTORS 2.2.3 THE DOT PRODUCT Solution If F and D are the force and displacement vectors, then the work done is W = F · D = | F || D | cos 35 = (70)(100) cos 35 5734 N · m = 5734 J . Assoc. Prof. Nguyen Ngoc Hai CALCULUS II VECTORS AND THE GEOMETRY OF SPACE
2.3 THE CROSS PRODUCT Definition 3.1 If a = h a 1 , a 2 , a 3 i and b = h b 1 , b 2 , b 3 i , then the cross product a × b of a and b is the vector a × b = h a 2 b 3 - a 3 b 2 , a 3 b 1 - a 1 b 3 , a 1 b 2 - a 2 b 1 i Cross product is also called the vector product . a × b is defined only when a and b are three-dimensional vectors . Assoc. Prof. Nguyen Ngoc Hai CALCULUS II VECTORS AND THE GEOMETRY OF SPACE
2.3 THE CROSS PRODUCT A determinant of order 2 is defined by a b c d = ad - bc . A determinant of order 3 can be defined in terms of second-order determinants as follows: a 1 a 2 a 3 b 1 b 2 b 3 c 1 c 2 c 3 = a 1 b 2 b 3 c 2 c 3 - a 2 b 1 b 3 c 1 c 3 + a 3 b 1 b 2 c 1 c 2 . Assoc. Prof. Nguyen Ngoc Hai CALCULUS II VECTORS AND THE GEOMETRY OF SPACE
2.3 THE CROSS PRODUCT Then the cross product of the vectors a = a 1 i + a 2 j + a 3 k and b = b 1 i + b 2 j + b 3 k is a × b = a 2 a 3 b 2 b 3 i - a 1 a 3 b 1 b 3 j + a 1 a 2 b 1 b 2 k = i j k a 1 a 2 a 3 b 1 b 2 b 3 . Assoc. Prof. Nguyen Ngoc Hai CALCULUS II VECTORS AND THE GEOMETRY OF SPACE
2.3 THE CROSS PRODUCT Example 3.1 Show that a × a = 0 for any vector a in R 3 . Solution If a = h a 1 , a 2 , a 3 i , then a × a = i j k a 1 a 2 a 3 a 1 a 2 a 3 = ( a 2 a 3 - a 3 a 2 ) i + ( a 3 a 1 - a 1 a 3 ) j + ( a 1 a 2 - a 2 a 1 ) k = 0 . Assoc. Prof. Nguyen Ngoc Hai CALCULUS II VECTORS AND THE GEOMETRY OF SPACE
2.3 THE CROSS PRODUCT Example 3.2 Show that i × j = k j × k = i k × i = j j × i = - k k × j = - i i × k = - j . Assoc. Prof. Nguyen Ngoc Hai CALCULUS II VECTORS AND THE GEOMETRY OF SPACE
2.3 THE CROSS PRODUCT Theorem 3.1 The vector a × b is orthogonal to both a and b .
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