Explanation Given: The point Q ( − 5 , 2 , 9 ) and the line r ( t ) = 〈 5 t + 7 , 2 − t , 12 t + 4 〉 . Formula used: Suppose the point is Q and the line is r = r 0 + t v . The distance d is | P Q ⇀ × v | | v | where P is the point on the line r . The cross product of two vectors u = 〈 a , b , c 〉 and v = 〈 x , y , z 〉 is given by u × v = | i j k a b c x y z | . Calculation: Consider the line r ( t ) = 〈 5 t + 7 , 2 − t , 12 t + 4 〉 and simplify as follows, r ( t ) = 〈 5 t + 7 , 2 − t , 12 t + 4 〉 = 〈 7 , 2 , 4 〉 + 〈 5 t , − t , 12 t 〉 = 〈 7 , 2 , 4 〉 + t 〈 5 , − 1 , 12 〉 Compare the above equation with the line equation r = r 0 + t v . Observe that the direction vector of the line v is 〈 5 , − 1 , 12 〉 and the position vector of the point P is 〈 7 , 2 , 4 〉 The vector P Q ⇀ is obtained as follows, P Q ⇀ = O Q ⇀ − O P ⇀ = 〈 − 5 , 2 , 9 〉 − 〈 7 , 2 , 4 〉 = 〈 − 5 − 7 , 2 − 2 , 9 − 4 〉 = 〈 − 12 , 0 , 5 〉 Thus, the vector P Q ⇀ is 〈 − 12 , 0 , 5 〉

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Chapter 11.5, Problem 81E

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To find: The least distance from the point Q(−5,2,9) and the line r(t)=〈5t+7,2−t,12t+4〉 .

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Chapter 11.5, Problem 80E

Chapter 11.5, Problem 82E

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The least distance from the point Q ( − 5 , 2 , 9 ) and the line r ( t ) = 〈 5 t + 7 , 2 − t , 12 t + 4 〉 .

Solution Summary: The author calculates the least distance between the point Q(-5,2,9) and the line r (t )=langle 5t+7,2-t,12

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To find: The least distance from the point Q(−5,2,9) and the line r(t)=〈5t+7,2−t,12t+4〉 .

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Chapter 11 Solutions