JEE Main & Advanced
Mathematics
Vector Algebra
Question Bank
Critical Thinking
question_answer
The points \[O,\,A,\,B,\,C,\,D\] are such that \[\overrightarrow{OA}=\mathbf{a},\] \[\overrightarrow{OB}=\mathbf{b},\,\] \[\overrightarrow{OC}=2\mathbf{a}+3\mathbf{b}\] and \[\overrightarrow{OD}=\mathbf{a}-2\mathbf{b}.\] If \[|\mathbf{a}|\,=3\,|\mathbf{b}|,\] then the angle between \[\overrightarrow{BD}\] and \[\overrightarrow{AC}\] is
A)\[\frac{\pi }{3}\]
B)\[\frac{\pi }{4}\]
C)\[\frac{\pi }{6}\]
D)None of these
Correct Answer:
D
Solution :
We have \[\overrightarrow{BD}=\overrightarrow{OD}-\overrightarrow{OB}=\mathbf{a}-2\mathbf{b}-\mathbf{b}=\mathbf{a}-3\mathbf{b}\] and \[\overrightarrow{AC}=\overrightarrow{OC}-\overrightarrow{OA}=2\mathbf{a}+3\mathbf{b}-\mathbf{a}=\mathbf{a}+3\mathbf{b}.\]
Let \[\theta \] be the angle between \[\overrightarrow{BD}\] and \[\overrightarrow{AC}.\]
Then \[\cos \theta =\frac{\overrightarrow{BD}\,.\,\overrightarrow{AC}}{|\overrightarrow{BD}|\,|\overrightarrow{AC}|}=\frac{|\mathbf{a}{{|}^{2}}-9|\mathbf{b}{{|}^{2}}}{|\overrightarrow{BD}|\,|\overrightarrow{AC}|}\]