Question:
Let \(O\) be the origin in 3D space. If the vertices \(A(2,p,-3),B(q,-2,5)\), and \(C(-5,1,r)\) of a triangle satisfy the vector relation \(\overline{PA}+\overline{PB}+\overline{PC}=3\overline{PO}\) for any point \(P\) in space, then the values of \((p,q,r)\) respectively are ...
Let \(O\) be the origin in 3D space. If the vertices \(A(2,p,-3),B(q,-2,5)\), and \(C(-5,1,r)\) of a triangle satisfy the vector relation \(\overline{PA}+\overline{PB}+\overline{PC}=3\overline{PO}\) for any point \(P\) in space, then the values of \((p,q,r)\) respectively are ...
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The relation PA + PB + PC = 3PO means the centroid of the triangle is the origin.
Updated On: Nov 12, 2026
- \((1,3,2)\)
- \((1,3,-2)\)
- \((1,2,3)\)
- \((1,-2,3)\)
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The Correct Option is B
Solution and Explanation
Step 1: Understanding the Concept
For the centroid \(G\) of triangle \(ABC\), \(\overrightarrow{PA}+\overrightarrow{PB}+\overrightarrow{PC}=3\overrightarrow{PG}\) for any point \(P\).
Step 2: Compare
The given relation means \(G\) coincides with \(O\), so the sum of the position vectors of \(A,B,C\) is zero.
Step 3: Equate components
x: \(2+q-5=0\), so \(q=3\).
y: \(p-2+1=0\), so \(p=1\).
z: \(-3+5+r=0\), so \(r=-2\).
Step 4: Compare the options
\((p,q,r)=(1,3,-2)\).
Final Answer:
The values are \((p,q,r)=(1,3,-2)\). \[ \boxed{\text{(B) }(1,3,-2)} \]
For the centroid \(G\) of triangle \(ABC\), \(\overrightarrow{PA}+\overrightarrow{PB}+\overrightarrow{PC}=3\overrightarrow{PG}\) for any point \(P\).
Step 2: Compare
The given relation means \(G\) coincides with \(O\), so the sum of the position vectors of \(A,B,C\) is zero.
Step 3: Equate components
x: \(2+q-5=0\), so \(q=3\).
y: \(p-2+1=0\), so \(p=1\).
z: \(-3+5+r=0\), so \(r=-2\).
Step 4: Compare the options
\((p,q,r)=(1,3,-2)\).
Final Answer:
The values are \((p,q,r)=(1,3,-2)\). \[ \boxed{\text{(B) }(1,3,-2)} \]
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