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Question
If `vec"PO" + vec"OQ" = vec"QO" + vec"OR"`, prove that the points P, Q, R are collinear
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Solution
Let P, Q, R be the given points.
Let O be the origin.
Then the position vectors of P, Q, R are `vec"OP", vec"OQ"` and `vec"OR"`
Given `vec"PO" + "OQ" = vec"QO" + vec"OR"` .......(1)
From the figure
`vec"PO" + "OQ" = vec"PQ"` ........(2)
`vec"QO" + "OR" = vec"QR"` ........(3)
Using equation (1), (2) and (3)
`vec"PQ" = vec"QR"`
`vec"PQ"` and `vec"QR"` are parallel vectors and are in the same direction, Q is a common point.
∴ P, Q, R lie on a straight line.
Hence, P, Q, R are collinear.
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