Let a , b be two vectors, and let P, Q and R be the points with position vectors a , b and a + b , respectively, with respect to the origin O. If | a + b | = 21 , | a - b | = 3, and a and ( a - b ) are perpendicular to each other, then the area of the triangle OPR is
C. 3 3 2
Given | a + b | = 21 and | a - b | = 3. Squaring both equations: | a |^2 + | b |^2 + 2 a b = 21 | a |^2 + | b |^2 - 2 a b = 9 Subtracting the second equation from the first gives: 4 a b = 12 a b = 3 Adding the two equations gives: 2(| a |^2 + | b |^2) = 30 | a |^2 + | b |^2 = 15 Since a and ( a - b ) are perpendicular, their dot product is zero: a ( a - b ) = 0 | a |^2 - a b = 0 | a |^2 = a b = 3 Substituting | a |^2 = 3 into | a |^2 + | b |^2 = 15 gives: 3 + | b |^2 = 15 | b |^2 = 12 Using Lagrange's identity to find | a b |: | a b |^2 = | a |^2 | b |^2 - ( a b )^2 | a b |^2 = (3)(12) - (3)^2 = 36 - 9 = 27 | a b | = 27 = 3 3 The position vectors of P and R are a and a + b respectively. The area of triangle OPR is: 1 2 | OP OR | = 1 2 | a ( a + b )| = 1 2 | a a + a b | = 1 2 | a b | Substituting the value of | a b |: Area = 3 3 2 Answer: 3 3 2
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