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Show that the vectors a =3hati - 2hatj+h...

Show that the vectors `a =3hati - 2hatj+hatk, b=hati - 3hatj+5hatk` and `c=2hatj+hatj-4hatk` form a right angled triangle.

Text Solution

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We have `b+c = (hati-3hatj+5hatk)+(2hatj+hatj-4hatk) = 3hati - 2hatj+hatk = a`
Hence, a,b,c are coplanar.
Also, we observe that no two of theses vectors are parallel.
Further, `a.c=(3hati - 2hatj+hatk).(2hati + hatj-4hatk) =0`
Dot product of two non-zero vectors is zero. Hence, they are perpendiuclar
so they form a right angled triangle.
`|a| = sqrt(9+4+1) = sqrt14`,
`|b|=sqrt(1+9+25)= sqrt35`
`|c| = sqrt(4+1+16)=sqrt21`
`rArr sqrt(a^2+c^2) = sqrtb^2`
.
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