The points O,A,B,C,D are such that ¯¯¯¯¯¯¯¯OA=a,¯¯¯¯¯¯¯¯OB=b,¯¯¯¯¯¯¯¯OC=2a+3b and ¯¯¯¯¯¯¯¯¯OD=a−2b. If |a|=3|b|, then the angle between ¯¯¯¯¯¯¯¯¯BD ,¯¯¯¯¯¯¯¯AC is
A
π3
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B
π4
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C
π6
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D
noneofthese
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Solution
The correct option is Dnoneofthese We have ¯¯¯¯¯¯¯¯¯BD=¯¯¯¯¯¯¯¯¯OD−¯¯¯¯¯¯¯¯OB=a−2b−b=a−3b and ¯¯¯¯¯¯¯¯AC=¯¯¯¯¯¯¯¯OC−¯¯¯¯¯¯¯¯OA=2a+3b−a=a+3b. Let θbe the angle between ¯¯¯¯¯¯¯¯¯BD and ¯¯¯¯¯¯¯¯AC. Then cosθ=¯¯¯¯¯¯¯BD.¯¯¯¯¯¯¯AC|¯¯¯¯¯¯¯BD||¯¯¯¯¯¯¯AC|=|a|2−9|b|2|¯¯¯¯¯¯¯BD||¯¯¯¯¯¯¯AC|9|b|2−9|b|2|¯¯¯¯¯¯¯BD||¯¯¯¯¯¯¯AC|,(∵|a|=3|b|)⇒cosθ=0∘⇒θ=π2.