Question

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 \(\left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}\right)\) (Hint: Let \(O\) be the origin and let \(M\) be the midpoint of \(P Q\). Draw a picture and show that $$\left.\overrightarrow{O M}=\overrightarrow{O P}+\frac{1}{2} \overrightarrow{P Q}=\overrightarrow{O P}+\frac{1}{2}(\overrightarrow{O Q}-\overrightarrow{O P}) \cdot\right)$$

Short answer

Question: Prove that the midpoint of the line segment joining two points P(x1, y1) and Q(x2, y2) is the point \((\frac{x1+x2}{2}, \frac{y1+y2}{2})\) using vectors.

Step-by-step solution

Define Points and Vectors

Let P(x1, y1) and Q(x2, y2) be two points in the plane and define the midpoint M as \((\frac{x1+x2}{2}, \frac{y1+y2}{2})\). Let O(0, 0) be the origin. We need to find the vectors \(\overrightarrow{OP}\), \(\overrightarrow{OQ}\), and \(\overrightarrow{PQ}\) and then use them to find the vector \(\overrightarrow{OM}\).

Find Vectors \(\overrightarrow{OP}\) and \(\overrightarrow{OQ}\)

To find the vectors \(\overrightarrow{OP}\) and \(\overrightarrow{OQ}\), subtract the coordinates of the origin from the coordinates of the respective points. \(\overrightarrow{OP}= =\) \(\overrightarrow{OQ}= =\)

Find Vector \(\overrightarrow{PQ}\)

To find the vector \(\overrightarrow{PQ}\), subtract the coordinates of P from the coordinates of Q. \(\overrightarrow{PQ} = \overrightarrow{OQ} - \overrightarrow{OP} =\)

Express the Vector \(\overrightarrow{OM}\) in terms of \(\overrightarrow{OP}\) and \(\overrightarrow{PQ}\)

Using the Hint equation provided, we can express the vector \(\overrightarrow{OM}\) as: $$\overrightarrow{OM} = \overrightarrow{OP} + \frac{1}{2} \overrightarrow{PQ}$$ $$\overrightarrow{OM} = + \frac{1}{2}()$$

Simplify the Vector \(\overrightarrow{OM}\)

Simplifying the vector \(\overrightarrow{OM}\), we obtain: $$\overrightarrow{OM} = + <\frac{1}{2}(x2-x1), \frac{1}{2}(y2-y1)>$$ $$\overrightarrow{OM} = $$ $$\overrightarrow{OM} = <\frac{x1+x2}{2}, \frac{y1+y2}{2}>$$

Final step: Compare the Coordinates of M and \(\overrightarrow{OM}\)

The coordinates of M are \((\frac{x1+x2}{2}, \frac{y1+y2}{2})\), and we have found the vector \(\overrightarrow{OM}\) to be \(<\frac{x1+x2}{2}, \frac{y1+y2}{2}>\). Since the coordinates match, we have shown that the midpoint of the line segment joining P(x1, y1) and Q(x2, y2) is the point \((\frac{x1+x2}{2}, \frac{y1+y2}{2})\).

Key concepts

Vectors in Plane

In the realm of mathematics, vectors play a pivotal role, especially when dealing with plane geometry. A vector in the plane is a directed line segment that has both magnitude (also known as length or norm) and direction. Each vector can be defined by its horizontal (x) and vertical (y) components, which are real numbers representing how far the vector moves in each direction.

For instance, the vector \(\overrightarrow{AB}\) is described as moving from point A to point B in the plane. This vector can be noted in component form as \(\overrightarrow{AB} = \), where \(A_x\) and \(A_y\) are the coordinates of the starting point A, and \(B_x\) and \(B_y\) are the coordinates of the endpoint B. Vectors are essential tools for conveying the relationship between points in the plane as they encapsulate both the distance and the direction from one point to another, which is crucial in many fields such as physics, engineering, and computer science.

Coordinate Geometry

Coordinate geometry, often known as analytic geometry, is the study of geometry using a coordinate system. This method fuses algebraic equations with geometric curves and surfaces to work with figures and points in the plane. By using a pair of numerical coordinates—typically denoted as \(x\) and \(y\)—each point in the plane can be uniquely identified.

The coordinate system most commonly used is the Cartesian coordinate system, where the horizontal axis is labeled the x-axis and the vertical axis is labeled the y-axis. The intersection of these axes is called the origin, often labeled as \(O(0, 0)\). The location of any point \(P(x, y)\) can then be described in relation to the origin and is fundamental in solving problems like finding the midpoint of a line segment, calculating the slope of a line, or determining the distance between two points. These operations are important not only in mathematics but also in various applications like navigation, computer graphics, and spatial analysis.

Vector Operations

Vector operations are fundamental for many calculations in mathematics and physics, including finding the midpoint of a line segment. Among the most common vector operations are vector addition, subtraction, scalar multiplication, and dot product.

Vector addition consists of adding the corresponding components of two vectors. If we have vectors \(\overrightarrow{AB}\) and \(\overrightarrow{BC}\), where \(\overrightarrow{AB}=\) and \(\overrightarrow{BC}=\), their sum \(\overrightarrow{AC}=\) can be found directly by adding the respective x and y components.

Scalar multiplication involves multiplying each component of a vector by a scalar (a real number). If a vector \(\overrightarrow{AB}\) and a scalar \(k\) are given, the product \(k\overrightarrow{AB}\) is a vector that has the same direction as \(\overrightarrow{AB}\) but a magnitude that is scaled by \(k\). Vector subtraction and dot product follow similar principles, combining vectors in ways that reflect both geometry and algebra. This amalgamation makes vector operations incredibly versatile and valuable when working with multiple geometric entities in the plane.