Problem 30 - Precalculus with Calculus Previews

Question

Use the distance formula to determine whether the given points are collinear. $$ P_{1}(1,2,-1), P_{2}(0,3,2), P_{3}(1,1,-3) $$

Step-by-Step Solution

Answer

The points are not collinear.

Step 1: Recall the Distance Formula

To determine if the points are collinear, we first need to find the distance between each pair of points. The distance formula to calculate the distance between two points \( P(a,b,c) \) and \( Q(x,y,z) \) in three-dimensional space is given by:\[D = \sqrt{(x-a)^2 + (y-b)^2 + (z-c)^2}\]

Step 2: Calculate Distance D(P₁, P₂)

Substitute the given coordinates of points \( P_{1}(1,2,-1) \) and \( P_{2}(0,3,2) \) into the distance formula:\[D(P_{1}, P_{2}) = \sqrt{(0-1)^2 + (3-2)^2 + (2-(-1))^2} \= \sqrt{(-1)^2 + (1)^2 + (3)^2}= \sqrt{1 + 1 + 9} = \sqrt{11}\]

Step 3: Calculate Distance D(P₂, P₃)

Use the distance formula for points \( P_{2}(0,3,2) \) and \( P_{3}(1,1,-3) \):\[D(P_{2}, P_{3}) = \sqrt{(1-0)^2 + (1-3)^2 + ((-3)-2)^2} \= \sqrt{(1)^2 + (-2)^2 + (-5)^2}= \sqrt{1 + 4 + 25} = \sqrt{30}\]

Step 4: Calculate Distance D(P₁, P₃)

Similarly, compute the distance between points \( P_{1}(1,2,-1) \) and \( P_{3}(1,1,-3) \):\[D(P_{1}, P_{3}) = \sqrt{(1-1)^2 + (1-2)^2 + ((-3)-(-1))^2} \= \sqrt{(0)^2 + (-1)^2 + (-2)^2}= \sqrt{0 + 1 + 4} = \sqrt{5}\]

Step 5: Check Collinearity

For the points to be collinear, the sum of the distances between two pairs of points should equal the distance of the third pair. We need to check if:\( D(P_{1}, P_{2}) + D(P_{2}, P_{3}) = D(P_{1}, P_{3}) \)\(\sqrt{11} + \sqrt{30} eq \sqrt{5} \\)Therefore, the points are not collinear as the distances do not satisfy this condition.

Key Concepts

Collinear Points

Collinear points are points that lie on the same straight line. To check if multiple points are collinear, we use the concept of distance. Basically, if you can draw a straight line through all the points without lifting your pen, they're collinear.

In our exercise, the objective was to determine if the points \(P_1(1,2,-1)\), \(P_2(0,3,2)\), and \(P_3(1,1,-3)\) lie on the same line. One reliable way to check this is by using the distance formula. The idea is to add up the distances between two pairs of points. If the sum matches the distance of the third pair, then the points are collinear.

The calculation in the solution shows that the sum of the calculations from \(D(P_1, P_2)\) and \(D(P_2, P_3)\) does not equal \(D(P_1, P_3)\). Hence, the points are not collinear.

Three-Dimensional Geometry

In three-dimensional geometry, we're dealing with shapes and spaces in three dimensions - usually defined by the axes \(x\), \(y\), and \(z\). Points in this space are labeled with three coordinates to represent their positions. For example, a point \((x, y, z)\) would have specific values for \(x\), \(y\), and \(z\) that place it somewhere in 3D space.

This becomes more complex compared to two-dimensional space, as we have one extra dimension to consider. When working with three-dimensional problems, we use spatial reasoning to visualize points, lines, and planes. Calculating distances, like in this exercise, helps us determine relationships between these points, such as whether they are collinear.

Visualizing these points on a 3D

Distance Calculation

The distance formula is essential in geometry for finding the straight-line distance between two points. In three-dimensional geometry, this formula is extended to consider the \(x\), \(y\), and \(z\) coordinates. The formula is:

• \[D = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2 + (z_2-z_1)^2}\]

This allows for the calculation of the shortest path connecting two points in a 3D environment. It's an adaptation of the Pythagorean theorem extended into three dimensions.

In the exercise, we used this formula to find the distances between the points \(P_1\), \(P_2\), and \(P_3\). Each calculation involves plugging in the coordinates of two points and solving the equation step-by-step. After comparing calculated distances, we uncover the relationship between these points in space. Understanding distance calculation is a fundamental skill in solving geometry problems involving measurement in a three-dimensional setting.