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Question

The points O,A,B,C,D are such that ¯¯¯¯¯¯¯¯OA=a,¯¯¯¯¯¯¯¯OB=b,¯¯¯¯¯¯¯¯OC=2a+3b and ¯¯¯¯¯¯¯¯¯OD=a2b. 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
none of these
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Solution

The correct option is D none of these
We have ¯¯¯¯¯¯¯¯¯BD=¯¯¯¯¯¯¯¯¯OD¯¯¯¯¯¯¯¯OB=a2bb=a3b and ¯¯¯¯¯¯¯¯AC=¯¯¯¯¯¯¯¯OC¯¯¯¯¯¯¯¯OA=2a+3ba=a+3b.
Let θ be the angle between ¯¯¯¯¯¯¯¯¯BD and ¯¯¯¯¯¯¯¯AC.
Then cosθ=¯¯¯¯¯¯¯BD.¯¯¯¯¯¯¯AC|¯¯¯¯¯¯¯BD||¯¯¯¯¯¯¯AC|=|a|29|b|2|¯¯¯¯¯¯¯BD||¯¯¯¯¯¯¯AC|9|b|29|b|2|¯¯¯¯¯¯¯BD||¯¯¯¯¯¯¯AC|,(|a|=3|b|)cosθ=0θ=π2.

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