A point P lies on the line joining A(1,2,3) and B(2,10,1). If z-coordinate of P is 7, what is the sum of other two coordinates?

Explanation

Let point P divide segment AB in the ratio \lambda:1. The z-coordinate is given by z = \frac{\lambda(1) + 1(3)}{\lambda + 1} = 7. Solving for \lambda: \lambda + 3 = 7\lambda + 7 \implies 6\lambda = -4 \implies \lambda = -\frac{2}{3}. Using \lambda, the x-coordinate is \frac{(-2/3)(2) + 1(1)}{-2/3 + 1} = \frac{-4/3 + 3/3}{1/3} = -1. The y-coordinate is \frac{(-2/3)(10) + 1(2)}{-2/3 + 1} = \frac{-20/3 + 6/3}{1/3} = -14. The sum of the other two coordinates is x + y = -1 - 14 = -15.

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