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Question

Show that the vectors 2^i^j+^k,^i3^j5^k and 3^i4^j4^k form the vertices of a right angled triangle .

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Solution

Let vectors 2^i^j+^k,^i3^j5^k and 3^i4^j4^k be position vectors of points A,B and C respectively.
i.e., OA=2^i^j+^k,OB=^i3^j5^k and OC=3^i4^j4^k
Now, vectors AB,BC, and AC represent the sides of ΔABC.
AB=(12)^i+(3+1)^j+(51)^k=^i2^j6^k
BC=(31)^i+(4+3)^j+(4+5)^k=2^i^j+^k
AC=(23)^i+(1+4)^j+(1+4)^k=^i+3^j+5^k
|AB|=(1)2+(2)2+(6)2=1+4+36=41
|BC|=22+(1)2+12=4+1+1=6
|AC|=(1)2+32+52=1+9+25=35
|BC|2+|AC|2=6+35=41=|AB|2
Hence, ΔABC is a right-angled triangle.

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