Question

Midpoint of a line segment Use vectors to show that the midpoint of the line segment joining \(P\left(x_{1}, y_{1}\right)\) and \(Q\left(x_{2}, y_{2}\right)\) is the point \(\left(\left(x_{1}+x_{2}\right) / 2,\left(y_{1}+y_{2}\right) / 2\right) .\) (Hint: Let \(O\) be the origin and let \(M\) be the midpoint of \(P Q\). Draw a picture and show that \(\overrightarrow{O M}=\overrightarrow{O P}+\frac{1}{2} \overrightarrow{P Q}=\overrightarrow{O P}+\frac{1}{2}(\overrightarrow{O Q}-\overrightarrow{O P})\).

Short answer

Answer: The coordinates of the midpoint are \(\left(\frac{x_{1}+x_{2}}{2},\frac{y_{1}+y_{2}}{2}\right)\).

Step-by-step solution

Represent Points with Vectors

Let's represent each point with the respective vectors: - Point \(P\left(x_{1}, y_{1}\right)\): \(\overrightarrow{O P}=\begin{bmatrix} x_1 \\ y_1 \end{bmatrix}\). - Point \(Q\left(x_{2}, y_{2}\right)\): \(\overrightarrow{O Q}=\begin{bmatrix} x_2 \\ y_2 \end{bmatrix}\). - Point \(M\left(x_m, y_m\right)\): \(\overrightarrow{O M}=\begin{bmatrix} x_m \\ y_m \end{bmatrix}\).

Calculate Vector PQ

We can calculate vector \(\overrightarrow{P Q}\) by subtracting the vector \(\overrightarrow{O P}\) from \(\overrightarrow{O Q}\): \(\overrightarrow{P Q}=\overrightarrow{O Q}-\overrightarrow{O P}\) So, \(\overrightarrow{P Q}=\begin{bmatrix} x_2-x_1 \\ y_2-y_1 \end{bmatrix}\).

Calculate the Sum of OP and half PQ

Now, let's calculate \(\overrightarrow{O P}+\frac{1}{2} \overrightarrow{P Q}\) using the given formula. It is given in the formula that \(\overrightarrow{O M}=\overrightarrow{O P}+\frac{1}{2} \overrightarrow{P Q}\). So, \(\begin{bmatrix} x_m \\ y_m \end{bmatrix} =\begin{bmatrix} x_1 \\ y_1 \end{bmatrix} + \frac{1}{2} \begin{bmatrix} x_2-x_1 \\ y_2-y_1 \end{bmatrix}\). Now, let's calculate the answer for both \(x_m\) and \(y_m\). - \(x_m= x_1 + \frac{1}{2}(x_2-x_1) = \frac{x_1 + x_2}{2}\) - \(y_m= y_1 + \frac{1}{2}(y_2-y_1) = \frac{y_1 + y_2}{2}\)

Conclude the Midpoint

Therefore, the midpoint \(M\) of the line segment joining \(P\left(x_{1}, y_{1}\right)\) and \(Q\left(x_{2}, y_{2}\right)\) is the point \(\left(\left(x_{1}+x_{2}\right) / 2,\left(y_{1}+y_{2}\right) / 2\right)\).

Key concepts

Midpoint

The midpoint of a line segment is a key concept in geometry. It is the point exactly halfway between two endpoints of a segment. In vector terms, the midpoint of a line segment connecting two points \(P(x_1, y_1)\) and \(Q(x_2, y_2)\) is calculated by averaging the coordinates of \(P\) and \(Q\). This is done by:

• Finding the average of the \(x\)-coordinates: \((x_1 + x_2) / 2\)
• Finding the average of the \(y\)-coordinates: \((y_1 + y_2) / 2\)

Applying these calculations, the midpoint \(M\) is given by the coordinates \(M((x_1 + x_2)/2, (y_1 + y_2)/2)\). This method simplifies determining the central point between two specific locations.

Line Segment

A line segment consists of two endpoints and all the points in between them. It is different from a line, which extends infinitely in both directions. In mathematics, we often represent a line segment by its endpoints, such as \(P(x_1, y_1)\) and \(Q(x_2, y_2)\). This exercise involves:

• Labeling and representing endpoints as vectors, such as \(\overrightarrow{OP}\) and \(\overrightarrow{OQ}\).
• Calculating the vector \(\overrightarrow{PQ}\), which connects the endpoints by subtracting one vector from the other: \(\overrightarrow{PQ} = \overrightarrow{OQ} - \overrightarrow{OP}\).

This concept helps us understand relationships and calculations related to distances and midpoints.

Origin

The origin plays a central role in vector problems. It is the point \((0, 0)\) in a coordinate system, acting as the starting reference point for vectors. In this exercise:

• We define vectors such as \(\overrightarrow{OP}\) and \(\overrightarrow{OQ}\) with the origin \(O\) as the initial point.
• The vector addition principle involves calculating vector \(\overrightarrow{OM}\), starting from the origin, using known vectors.

Utilizing the origin allows for consistency and simplicity in calculations involving vector addition and midpoint determination.

Vector Addition

Vector addition is a foundational aspect of understanding vector mechanics. It involves combining vectors to determine a resultant vector. Here's how it works in this context:

• To find \(\overrightarrow{OM}\), the formula \(\overrightarrow{OM} = \overrightarrow{OP} + \frac{1}{2} \overrightarrow{PQ}\) is used.
• This equation allows us to express \(OM\) as the sum of vector \(OP\) and half of vector \(PQ\).

By adding vectors, we ascertain both the direction and magnitude of resultant vectors, making calculations clear and systematic. This principle is crucial in determining points like midpoints effectively.