hwsoln2 - CHAPTER 10 HOMEWORK SOLUTIONS Section 10.4 2....

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Unformatted text preview: CHAPTER 10 HOMEWORK SOLUTIONS Section 10.4 2. Find the length and direction of A × B and B × A . Where A = 2 i + 3 j and B =- i + j A × B = i j k 2 3 0- 1 0 0 = 0 i +0 j +(2- (- 3)) k = 5 k ⇒ length=5 and direction= k . B × A =- ( A × B ) ⇒ length=5 and direction=- k . 8. Find the length and direction of A × B and B × A . Where A = 3 2 i- 1 2 j and B = i + j + 2 k A × B = i j k 3 2- 1 2 1 1 1 = (- 1 2 * 2- 1 * 1) i- ( 3 2 * 2- 1 * 1) j + ( 3 2 * 1- (- 1 2 * 1)) k =- 2 i- 2 j + 2 k ⇒ length= √ 4 + 4 + 4 = 2 √ 3 and direction=- 1 √ 3 i- 1 √ 3 j + 1 √ 3 k . B × A =- ( A × B ) ⇒ length=2 √ 3 and direction= 1 √ 3 i + 1 √ 3 j- 1 √ 3 k . 12. Sketch the vectors A = 2 i- j , B = i + 2 j , and A × B with all three vectors starting at the origin. A × B = i j k 2- 1 0 1 2 = 5 k 0.0 0.5 1.0 1.5 2.0 x- 1 1 2 y 2 4 z 1 2 CHAPTER 10 HOMEWORK SOLUTIONS 16. a) Find the area of the triangle determined by the points P (1 , 1 , 1), Q (2 , ,- 1), and R (3 ,- 1 , 1). Two vectors that connect the points:-→ PQ = (1 , , 2) and-→ PR = (2 ,- 2 , 0) The cross product-→ PQ ×-→ PR has magnitude equal to the parallelogram created by-→ PQ and-→ PR which is twice the area of the triangle given by P , Q , and R .-→ PQ ×-→ PR = i j k 1 2 2- 2 0 = 4 i + 4 j- 2 k Area of triangle = 1 2 |-→ PQ ×-→ PR | = 1 2 √ 16 + 16 + 4 = 3 b)Find a unit vector perpendicular to the plane PQR . The unit vector in the directions of-→ PQ ×-→ PR and--→ PQ ×-→ PR, namely ± ( 4 3 i + 4 3 j- 2 3 k ) 28. a)True: A · B = a 1 b 1 + a 2 b 2 + a 3 b 3 = b 1 a 1 + b 2 a 2 + b 3 a 3 = B · A b)True: A × B = i j k a 1 a 2 a 3 b 1 b 2 =- i j k b 1 b 2 b 3 a 1 a 2 a 3 =- B × A c)True: (- A ) × B = i j k- a 1- a 2- a 3 b 1 b 2 b 3 =- i j k a 1 a 2 a 3 b 1 b 2 b 3 =- ( A × B ) d)True: ( c A ) · B = ( ca 1 ) b 1 + ( ca 2 ) b 2 + ( ca 3 ) b 3 = a 1 ( cb 1 ) + a 2 ( cb 2 ) + a 3 ( cb 3 ) = A · ( c B ) = ca...
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This note was uploaded on 10/24/2010 for the course APPM 2350 at Colorado.

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hwsoln2 - CHAPTER 10 HOMEWORK SOLUTIONS Section 10.4 2....

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