Section Formula

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A point P divides the line segment joining A(1,2,3) and B(4,8,9) internally in the ratio 2:1; locate the position vector of P.

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The section formula for internal division states that a point dividing the segment from a to b in ratio m:n has position vector (m b + n a)/(m + n). This weighted-average principle is a recurring JEE Advanced tool for locating points along directed segments. With a = (1,2,3), b = (4,8,9), m = 2 and n = 1, compute (2 b + 1 a)/3 componentwise. The x-coordinate is (2(4) + 1(1))/3 = (8 + 1)/3 = 9/3 = 3. The y-coordinate is (2(8) + 1(2))/3 = (16 + 2)/3 = 18/3 = 6. The z-coordinate is (2(9) + 1(3))/3 = (18 + 3)/3 = 21/3 = 7. Hence P = 3i + 6j + 7k. Option 2i + 4j + 5k is the midpoint, using ratio 1:1 instead. Option 3i + 6j + 6k miscomputes the z-coordinate. Option 5i + 10j + 12k forgets to divide by (m + n). Since the ratio 2:1 places P twice as far from A as from B, P should sit nearer B, which the result confirms. Plausibility check: each coordinate of P lies strictly between the corresponding coordinates of A and B, as required for internal division.

Section Formula

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