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Question
The value of a, for which the points A,B,C with position vectors 2^i−^j+^k,^i−3^j−5^k,a^i−3^j+^k respectively are the vertices of a right angled triangle with C=π2 are:
The value of a, for which the points A,B,C with position vectors 2^i−^j+^k,^i−3^j−5^k,a^i−3^j+^k respectively are the vertices of a right angled triangle with C=π2 are:
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Solution
The correct options are
A 1
D 2
−−→BC=−−→OC−−−→OB
=a^i−3^j+^k−(^i−3^j−5^k)
=(a−1)^i+6^k
−−→CA=−−→OA−−−→OC
=2^i−^j+^k−(a^i−3^j+^k)
=(2−a)^i+2^j
−−→BC.−−→CA=[(a−1)^i+6^k].[(2−a)^i+2^j]
=(a−1)(2−a)+0
since ^i.^i=1,^i.^j=0
⇒(a−1)(2−a)=0
∴a=1,2
A 1
D 2
−−→BC=−−→OC−−−→OB
=a^i−3^j+^k−(^i−3^j−5^k)
=(a−1)^i+6^k
−−→CA=−−→OA−−−→OC
=2^i−^j+^k−(a^i−3^j+^k)
=(2−a)^i+2^j
−−→BC.−−→CA=[(a−1)^i+6^k].[(2−a)^i+2^j]
=(a−1)(2−a)+0
since ^i.^i=1,^i.^j=0
⇒(a−1)(2−a)=0
∴a=1,2
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