The harmonic conjugate of P(-9, 12, -15) with | vedclass

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(C) Let the point $P(-9, 12, -15)$ divide the line segment joining $A(1, -2, 3)$ and $B(-4, 5, -6)$ in the ratio $\lambda : 1$. Using the section form

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$\left(-\frac{7}{3}, \frac{8}{3}, -3\right)$

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(C) Let the point $P(-9, 12, -15)$ divide the line segment joining $A(1, -2, 3)$ and $B(-4, 5, -6)$ in the ratio $\lambda : 1$. Using the section formula,we have: $-9 = \frac{-4\lambda + 1}{\lambda + 1} \implies -9\lambda - 9 = -4\lambda + 1 \implies -5\lambda = 10 \implies \lambda = -2$. The point $P$ divides $AB$ externally in the ratio $2 : 1$. The harmonic conjugate of $P$ with respect to $AB$ is the point $Q$ that divides $AB$ internally in the same ratio $2 : 1$. Using the internal section formula: $Q = \left(\frac{2(-4) + 1(1)}{2+1}, \frac{2(5) + 1(-2)}{2+1}, \frac{2(-6) + 1(3)}{2+1}\right)$ $Q = \left(\frac{-8 + 1}{3}, \frac{10 - 2}{3}, \frac{-12 + 3}{3}\right)$ $Q = \left(-\frac{7}{3}, \frac{8}{3}, -3\right)$.

The harmonic conjugate of $P(-9, 12, -15)$ with respect to the line segment $AB$,where $A=(1, -2, 3)$ and $B=(-4, 5, -6)$ is

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