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Show that the points A,B and C with posi...

Show that the points A,B and C with position vectors `a=3hatj-4hatj-4hatk,b=2hati-hatj+hatk and c=hati-3hatj-5hatk` respectively, form the vertices of a right angled triangle.

Text Solution

Verified by Experts

We have `vec(AB)`=(position vector of B)-(position vector of A)
`=(3hati-hatj+hatk)-(3hati-4hatj-4hatk)`
`=(-hati+3hatj+5hatk)`
`vec(BC)`=(position vector of C)-(position vector of B)
`=(hati-3hatj-5hatk)-(2hati-hatj+hatk)=(-hati-2hatj-6hatk)`
`vec(CA)`=(position vector of A)-(position vector of C)
`=(2hati-hatj+hatk)`
`:.|vec(AB)|^(2)=35`
`|vec(BC)|^(2)=41`
And `|vec(CA)|^(2)=6`
Thus `|vec(AB)|^(2)+|vec(CA)|^(2)=|vec(BC)|^(2) rArrAB^(2)+CA^(2)=BC^(2)`
Hence `Delta` ABC is right angled at A.
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