Question

Given \(P_{1}(2,1,-2)\) and \(P_{2}(1,-2,0)\) Find the coordinates of the point \(P\) : a. \(\frac{1}{5}\) the way from \(P_{1}\) to \(P_{2}\) b. \(\frac{1}{4}\) the way from \(P_{2}\) to \(P_{1}\)

Short answer

(9/5, 4/5, -8/5) and (5/4, -5/4, -1/2)

Step-by-step solution

Understanding the Problem

We need to find a point that is a specific fraction of the distance from one point to another in a 3D coordinate system. This involves interpolation between points on each coordinate axis.

Determine Direction Vectors

To find the point along the line from \(P_1\) to \(P_2\), first determine the direction vector: \(d = P_2 - P_1 = (1 - 2, -2 - 1, 0 - (-2)) = (-1, -3, 2)\).

Point 1/5 of the Way From P1 to P2

Find point \(P\) which is \(\frac{1}{5}\) of the way along the direction vector: \[ P = P_1 + \frac{1}{5}d = (2, 1, -2) + \frac{1}{5}(-1, -3, 2) = \left(2 - \frac{1}{5}, 1 - \frac{3}{5}, -2 + \frac{2}{5}\right) \]Result: \(\left(\frac{9}{5}, \frac{4}{5}, -\frac{8}{5}\right)\).

Point 1/4 of the Way From P2 to P1

Now find point \(P\) which is \(\frac{1}{4}\) of the way back along the same direction vector starting from \(P_2\): \[ P = P_2 + \frac{1}{4}(-d) = (1, -2, 0) + \frac{1}{4}(1, 3, -2) = \left(1 + \frac{1}{4}, -2 + \frac{3}{4}, 0 - \frac{1}{2}\right) \]Result: \(\left(\frac{5}{4}, -\frac{5}{4}, -\frac{1}{2}\right)\).

Key concepts

Interpolation

Interpolation in 3D coordinate geometry involves finding a point that lies at a certain fractional distance between two known points. Imagine you're walking from one point to another. But instead of walking the whole distance, you stop at a certain fraction of the journey. That's what interpolation helps us find: that exact stop point.

In mathematical terms, when you interpolate, you calculate this intermediate point using vector math. If you have two points, say \(P_1\) and \(P_2\), you can find a point \(P\) that lies, for example, a fraction \(\frac{1}{n}\) between \(P_1\) and \(P_2\).

• If the fraction is \(\frac{1}{5}\), it means \(P\) is closer to \(P_1\), only \(\frac{1}{5}\) of the way from \(P_1\) to \(P_2\).
• Conversely, a fraction \(\frac{4}{5}\) means \(P\) is more towards \(P_2\) from \(P_1\).

Interpolation is a vital concept because it can be used not only in geometry but also in computer graphics, data analysis, and more where finding intermediate values is necessary.

Direction Vectors

Direction vectors help us see how one point moves in space towards another point. It's like having GPS coordinates that tell us the direction we need to move.

To find this direction, from \(P_1\) to \(P_2\), you subtract each coordinate of \(P_1\) from \(P_2\). In our problem:

• \(d = P_2 - P_1 = (1 - 2, -2 - 1, 0 - (-2)) = (-1, -3, 2)\)

This vector, \((-1, -3, 2)\), shows us the exact movement needed along each coordinate axis.

• "\(-1\)" in the x-direction tells us to move one unit left.
• "\(-3\)" in the y-direction means go down three units.
• "2" in the z-direction means go up two units.

By knowing the direction vector, you can easily interpolate any other point along this line, using fractions, just like in our examples.

3D Points

3D points describe locations in space using three coordinates, usually labeled as \(x\), \(y\), and \(z\). Each coordinate tells us how far the point is along each axis:

• \(x\) is the horizontal distance, or left-right position.
• \(y\) is the vertical distance, or up-down position.
• \(z\) is the depth or forward-backward position.

Our original points, \(P_1 = (2,1,-2)\) and \(P_2 = (1,-2,0)\), give us clear spots in this three-dimensional space. You can think of each point like a spot where lines from each axis intersect.

Many problems in geometry involve pinpointing such locations, interpolating between them, or even determining their directional relationships, much like mapping out a path from one spot in a 3D video game to another.