Problem 15 - Advanced Engineering Mathematics

Question

Find the vector \(P_{1} P_{2}\). Graph \(P_{1} P_{2}\) and its corresponding position vector. \(P_{1}(3,2), P_{2}(5,7)\)

Step-by-Step Solution

Answer

Vector \(P_1P_2 = (2, 5)\). It is graphed from \(P_1\) to \(P_2\).

Step 1: Understanding the Problem

We are given two points in a 2D plane, \(P_1(3,2)\) and \(P_2(5,7)\). Our task is to find the vector \(P_1P_2\), which describes the direction and magnitude from \(P_1\) to \(P_2\). We will also graph this vector and identify its position vector.

Step 2: Finding the Vector Components

To find the vector \(\vec{P_1P_2}\), use the formula for vector components: \(\vec{P_1P_2} = (x_2 - x_1, y_2 - y_1)\). Here, substitute \(x_1 = 3, y_1 = 2, x_2 = 5,\) and \(y_2 = 7\).

Step 3: Calculating the Components

Substituting the values, we get: \(\vec{P_1P_2} = (5 - 3, 7 - 2) = (2, 5)\). Thus, the vector \(\vec{P_1P_2}\) has components \(2\) and \(5\).

Step 4: Graphing the Vector and Position Vector

To graph \(\vec{P_1P_2}\), start at point \(P_1(3,2)\) and plot the end point at \(P_2(5,7)\). The vector \(\vec{P_1P_2}\) is directed from \(P_1\) to \(P_2\) with horizontal and vertical components of \(2\) and \(5\) respectively. The position vector of \(P_2\) can also be plotted from the origin (0,0) to point \(P_2\) with the same components.

Key Concepts

Vector Components

Vector components are crucial in understanding how vectors work. Imagine vectors as arrows with direction and magnitude. In two-dimensional space, each vector can be broken into two parts: horizontal and vertical components. These parts describe how far the vector moves along each axis.

To find these components, you use simple subtraction between the corresponding coordinates of the points. For the vector \(\vec{P_1P_2}\), the horizontal component is \(x_2 - x_1\), while the vertical is \(y_2 - y_1\).

Using the given points \(P_1(3,2)\) and \(P_2(5,7)\), the calculations are:

• Horizontal: \(5 - 3 = 2\)
• Vertical: \(7 - 2 = 5\)

This means the vector \(\vec{P_1P_2}\) has components \(2\) and \(5\). It signifies a movement 2 units right and 5 units up.

Position Vector

A position vector is a specific type of vector used to indicate the location of a point relative to the origin of a coordinate system. It's like a directed arrow going from the beginning of the axes to a specific point.

For any point, say \(P(x, y)\), its position vector is \(\vec{OP} = (x, y)\), with \(O\) being the origin \((0,0)\). It's essential in connecting geometric points to vector mathematics.

In our exercise, after calculating \(\vec{P_1P_2}\), we also identify the position vector of \(P_2\). The position vector \(\vec{OP_2}\) is shown directly from the origin to \(P_2(5,7)\), having the same components \(5\) and \(7\) that describe its position accurately.

Graphing Vectors

Graphing vectors helps visualize their direction and magnitude, making abstract numbers more concrete. Start by plotting the initial point, like \(P_1(3,2)\). From here, use the components to reach the endpoint, \(P_2(5,7)\).

This movement corresponds directly to drawing the vector \(\vec{P_1P_2}\), which directs from \(P_1\) to \(P_2\).

Represent it as a straight line with

• Length reflecting the magnitude of the components
• Arrow indicating direction

It's not just about labeling, but also appreciating how vectors traverse the plane.

Additionally, graphing the position vector from the origin to a point like \(P_2\) provides clear insights into its exact location and relationship with the start of the coordinate system.