Problem 52 - University Calculus: Early Transcendentals

Question

Use similar triangles to find the coordinates of the point \(Q\) that divides the segment from \(P_{1}\left(x_{1}, y_{1}, z_{1}\right)\) to \(P_{2}\left(x_{2}, y_{2}, z_{2}\right)\) into two lengths whose ratio is \(p / q=r\).

Step-by-Step Solution

Answer

The coordinates of point \( Q \) are \( \left(\frac{px_2 + qx_1}{p+q}, \frac{py_2 + qy_1}{p+q}, \frac{pz_2 + qz_1}{p+q}\right) \).

Step 1: Understand the problem

We need to find the coordinates of a point \( Q \) on the line segment joining points \( P_1(x_1, y_1, z_1) \) and \( P_2(x_2, y_2, z_2) \), such that the ratio of the lengths of \( P_1Q \) to \( QP_2 \) is \( \frac{p}{q} = r \). This is a problem involving similar triangles and the section formula.

Step 2: Apply section formula

The section formula in three dimensions states that a point \( Q(x, y, z) \) that divides a line segment joining points \( P_1(x_1, y_1, z_1) \) and \( P_2(x_2, y_2, z_2) \) in the ratio \( m:n \) has coordinates given by: \( (x, y, z) = \left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}, \frac{mz_2 + nz_1}{m+n}\right) \).

Step 3: Substitute into section formula

Substitute \( m = p \) and \( n = q \) into the section formula. The coordinates of point \( Q \) are: \( \left(\frac{px_2 + qx_1}{p+q}, \frac{py_2 + qy_1}{p+q}, \frac{pz_2 + qz_1}{p+q}\right) \).

Step 4: Verify the ratio condition

By substituting the coordinates of \( Q \) back into the section formula, verify that \( Q \) divides the line segment from \( P_1 \) to \( P_2 \) in the ratio \( \frac{p}{q} \). This can be checked by examining the linearity of the coordinates.

Key Concepts

Section Formula

The section formula is an invaluable tool in coordinate geometry which allows us to find a point on a line segment divided in a specific ratio. Imagine you have two points, say \( P_1(x_1, y_1, z_1) \) and \( P_2(x_2, y_2, z_2) \), and you want to find another point, \( Q(x, y, z) \), that divides this segment in a given ratio, \( m:n \). The section formula gives us exactly how to calculate the coordinates of point \( Q \).

• In two-dimensional space, the formula is \( (x, y) = \left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right) \).
• When we bring it to three-dimensional space, it becomes \( (x, y, z) = \left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}, \frac{mz_2 + nz_1}{m+n}\right) \).

This formula is derived by considering similar triangles and applying proportional division on each coordinate separately. By using it, you can easily find coordinates that would otherwise be difficult to compute manually.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, combines algebra and geometry to solve spatial problems with points, lines, and shapes. By assigning coordinates, we can use equations and algebraic techniques to explore geometric properties.

• Two main components are involved: the coordinate plane and algebraic expressions.
• Each point on the plane is defined by a set of numbers representing its position along the X, Y, and possibly Z-axes for three-dimensional problems.

We use coordinate geometry to find distances, midpoints, and angles, as well as to understand more complex figures and paths like curves and surfaces. It works by translating physical scenarios into mathematical models, making it easier to perform calculations and deductions. In our context, the section formula would be a way to deal with linear paths in this field, allowing us to utilize ratio and proportion concepts seamlessly.

Ratios and Proportions

Ratios and proportions are fundamental concepts in mathematics, offering a way to compare quantities. A ratio is simply a way to show how two numbers relate to each other by division, whereas a proportion implies that two ratios are equal.

• Ratios are expressed as \( a:b \) or \( \frac{a}{b} \).
• Proportions are equations that state that two ratios are equivalent, such as \( \frac{a}{b} = \frac{c}{d} \).

When we use the section formula in coordinate geometry, ratios determine the point's position along a line segment. For instance, if a point \( Q \) divides the line between \( P_1 \) and \( P_2 \) in the ratio \( p:q \), the measurements ensure \( P_1Q:QP_2 \) is precisely \( \frac{p}{q} \), maintaining proportional distances from each endpoint. Understanding and applying these concepts is crucial, as they offer a bridge between geometry and numerical analysis, allowing us to manipulate and solve geometric scenarios using algebraic techniques.