Find the co-ordinates of point P which is five sixth of the way from `A(-2,0,6)` and `B(-10,-6,-12)`

Answer

To find the coordinates of point P which is five-sixths of the way from point A to point B, we can use the section formula. The coordinates of points A and B are given as follows:

- A(-2, 0, 6)
- B(-10, -6, -12)

Since point P divides the line segment AB in the ratio of 5:1 (because it is five-sixths of the way from A to B), we can use the section formula which states:

If a point P divides the line segment joining points A(x1, y1, z1) and B(x2, y2, z2) in the ratio m:n, then the coordinates of point P are given by:

\[ P\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}, \frac{mz_2 + nz_1}{m+n}\right) \]

In our case, m = 5 and n = 1. Let's denote the coordinates of A and B:

- \(x_1 = -2\), \(y_1 = 0\), \(z_1 = 6\)
- \(x_2 = -10\), \(y_2 = -6\), \(z_2 = -12\)

Now we can substitute these values into the formula:

1. **Calculate the x-coordinate of P:**
\[ x = \frac{5(-10) + 1(-2)}{5 + 1} = \frac{-50 - 2}{6} = \frac{-52}{6} = -\frac{26}{3} \]

2. **Calculate the y-coordinate of P:**
\[ y = \frac{5(-6) + 1(0)}{5 + 1} = \frac{-30 + 0}{6} = \frac{-30}{6} = -5 \]

3. **Calculate the z-coordinate of P:**
\[ z = \frac{5(-12) + 1(6)}{5 + 1} = \frac{-60 + 6}{6} = \frac{-54}{6} = -9 \]

Now we can summarize the coordinates of point P:

\[ P\left(-\frac{26}{3}, -5, -9\right) \]

Thus, the coordinates of point P are:

\[ P\left(-\frac{26}{3}, -5, -9\right) \]