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Question
Prove that the points A=(1,2,3),B(3,4,7),C(−3,−2,−5) are collinear & find the ratio in which B divides AC.
Prove that the points A=(1,2,3),B(3,4,7),C(−3,−2,−5) are collinear & find the ratio in which B divides AC.
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Solution
The correct option is B 2:3
A(1,2,3),B(3,4,7),C(−3,−2,−5)−−→AB=2^i+2^j+4^k−−→BC−6^i−6^j−12^kIf points are collinear then |(−−→AB).(−−→BC)|=|−−→AB||−−→BC|
−−→AB.−−→AC=−12−12−24=−48|−−→AB.−−→AC|=48|−−→AB|.|−−→AC|=√24√216=2√24√24=48
Points are collinear|−−→AC|=|−4^i−4^j−8^k|=4√6|−−→AB|=2√6B divides AC in the ration 2:3
A(1,2,3),B(3,4,7),C(−3,−2,−5)
−−→AB=2^i+2^j+4^k
−−→BC−6^i−6^j−12^k
If points are collinear then |(−−→AB).(−−→BC)|=|−−→AB||−−→BC|
−−→AB.−−→AC=−12−12−24=−48
|−−→AB.−−→AC|=48
|−−→AB|.|−−→AC|=√24√216=2√24√24=48
Points are collinear
|−−→AC|=|−4^i−4^j−8^k|=4√6
|−−→AB|=2√6
B divides AC in the ration 2:3
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