Question

Let \(P_{1}\left(x_{1}, y_{1}, z_{1}\right)\) and \(P_{2}\left(x_{2}, y_{2}, z_{2}\right)\) be two points with vectors \(\mathbf{p}_{1}\) and \(\mathbf{p}_{2}\), respectively. If \(r\) and \(s\) are positive integers, show that the point \(P\) lying \(\frac{r}{r+s}\) the way from \(P_{1}\) to \(P_{2}\) has vector $$\mathbf{p}=\left(\frac{s}{r+s}\right) \mathbf{p}_{1}+\left(\frac{r}{r+s}\right) \mathbf{p}_{2}$$

Short answer

Point \( P \) lies \( \frac{r}{r+s} \) the way from \( P_1 \) to \( P_2 \) with vector \( \mathbf{p} = \left( \frac{s}{r+s} \right) \mathbf{p}_1 + \left( \frac{r}{r+s} \right) \mathbf{p}_2. \)

Step-by-step solution

Understand the Concept of Section Formula

The section formula in 3D geometry gives the coordinates of a point dividing the line segment joining two points, say \(P_1(x_1, y_1, z_1)\) and \(P_2(x_2, y_2, z_2)\), in a given ratio \(m:n\). When a point \( P \) divides the segment internally in the ratio \(r:s\), the coordinates of \( P \) are given by: \(\left( \frac{sx_1 + rx_2}{r+s}, \frac{sy_1 + ry_2}{r+s}, \frac{sz_1 + rz_2}{r+s} \right).\) This formula can be extended to vectors, which we will use here.

Apply the Section Formula to Vectors

Assume that \( \mathbf{p}_1 = \langle x_1, y_1, z_1 \rangle \) and \( \mathbf{p}_2 = \langle x_2, y_2, z_2 \rangle \). According to the section formula, the vector \( \mathbf{p} \) that represents the point dividing the segment in the ratio \(r:s\) toward \(P_2\) is given by: \(\mathbf{p} = \left(\frac{sx_1 + rx_2}{r+s}, \frac{sy_1 + ry_2}{r+s}, \frac{sz_1 + rz_2}{r+s}\right).\)

Rewrite Using Vector Notation

Express the vector \( \mathbf{p} \) as a combination of \( \mathbf{p}_1 \) and \( \mathbf{p}_2 \) directly:\[\mathbf{p} = \left( \frac{s}{r+s} \right) \mathbf{p}_1 + \left( \frac{r}{r+s} \right) \mathbf{p}_2.\]This forms a weighted sum where \( \mathbf{p}_1 \) is weighted by \( \frac{s}{r+s} \) and \( \mathbf{p}_2 \) by \( \frac{r}{r+s} \), aligning with the internal division at \( \frac{r}{r+s} \) the way from \( P_1 \) to \( P_2 \).

Key concepts

Vectors

Vectors are essential in mathematics, especially in representing points or positions in a space. They have both magnitude and direction. In 3D geometry, a vector can be thought of as an arrow pointing from one point to another, capturing both the distance and the directional aspect of this path. To define a vector in 3D space, we use coordinates based on the x, y, and z axes, for example, \( \mathbf{p} = \langle x, y, z \rangle \).When working with vectors, important operations include addition, subtraction, and multiplication. For example:

• Addition: Combining two vectors to find their resultant.
• Scalar Multiplication: Scaling a vector by a real number, which changes its magnitude but not its direction.

Understanding vectors introduces us to the concept of linear combinations, which are expressions like \( r \mathbf{p}_1 + s \mathbf{p}_2 \), where \( r \) and \( s \) are scalars. This lays the groundwork for tools like the section formula used in exercises involving points and lines in three dimensions.

Internal Division

Internal division refers to a scenario where a line segment is divided into two parts by a point lying within the segment. This is a crucial concept in geometry as it enables us to locate precise points that subdivide a line segment in specified ratios.The section formula is a mathematical formula used for finding the coordinates of the point dividing the line segment internally between two given points, \( P_1\left(x_1, y_1, z_1\right) \) and \( P_2\left(x_2, y_2, z_2\right) \). If the division ratio is given by \( r:s \), the point \( P \) has the coordinates:\[\left( \frac{sx_1 + rx_2}{r+s}, \frac{sy_1 + ry_2}{r+s}, \frac{sz_1 + rz_2}{r+s} \right)\]This formula is a key to not only finding the coordinates but also aids in understanding the proportional lengths and areas in geometric shapes formed by such divisions.

Weighted Sum

The concept of a weighted sum is an extension of ordinary sum whereby each component of the addition is multiplied by a coefficient that reflects its relative importance or size. This is particularly valuable when dealing with vectors as it succinctly represents the linear combination of vectors.In the context of vectors and the section formula, a weighted sum involves these fractions or weights: \( \frac{s}{r+s} \) and \( \frac{r}{r+s} \). Thus, given vectors \( \mathbf{p}_1 \) and \( \mathbf{p}_2 \), and weights based on the ratio \( r:s \), the vector \( \mathbf{p} \) that divides the line segment can be expressed as:\[\mathbf{p} = \left( \frac{s}{r+s} \right) \mathbf{p}_1 + \left( \frac{r}{r+s} \right) \mathbf{p}_2\]This form emphasizes how each vector contributes to the final position: \( \mathbf{p}_1 \) and \( \mathbf{p}_2 \) are scaled according to their required proportion, elegantly expressing the new vector as a mixture of the two original vectors. The weighted sum is essential not only in geometry but also in fields like machine learning, where it helps in combining various inputs to produce an output.