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If the origin is the centroid of the tri...

If the origin is the centroid of the triangle whose vertices are A (2,p,-3), B(q,-2,5) and C(-5,1,r), then find the values of p,q and r.

Text Solution

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Let `bar(a),bar(b),bar(c)` be the position vectors of the vertices A,B,C of the triangle respectively.
Then `bar(a)=2hat(i)+phat(j)-3hat(k),bar(b)=qhat(i)-2hat(j)+5hat(i)+hat(j)+rhat(k)`
Since origin is the centroid of triangle ABC, by centroid formula, we get
`bar(0)=(bar(a)+bar(b)+bar(c))/(3)`
`:.bar(a)+bar(b)+bar(c)=bar(0)`
`:.(2hat(i)+phat(j)-3hat(k))+(qhat(i)-2hat(j)+5hat(k))+(-5hat(i)+hat(j)+rhat(k))=bar(0)`
`:.(2+q-5)hat(i)+(p-2+1)hat(j)+(-3+5+r)hat(k)=0*hat(i)+0*hat(j)+0*hat(k)`
`:.` by equality of vectors,
`q-3=0,p-1=0` and `r+2=0`
`:.q=3,p=1` and `r=-2` Hence, `p=1,q=3` and `r=-2`.
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