Problem 17 - Advanced Engineering Mathematics

Question

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

Step-by-Step Solution

Answer

The vector \(P_1P_2\) is \(\langle 2, 2 \rangle\).

Step 1: Determine the Components of P1P2

To find the vector from point \(P_1 = (3, 3)\) to point \(P_2 = (5, 5)\), subtract the coordinates of \(P_1\) from those of \(P_2\). This will give you the vector's direction and magnitude. Compute: \((5 - 3, 5 - 3) = (2, 2)\). Thus, the vector \(P_1P_2\) is \(\langle 2, 2 \rangle\).

Step 2: Plot the Points on a Graph

Draw a coordinate plane and plot the points \(P_1(3, 3)\) and \(P_2(5, 5)\). These points are the tail and head of your vector \(P_1P_2\), showing the vector's origin and destination.

Step 3: Graph the Vector P1P2

Use the points \(P_1\) and \(P_2\) to draw the vector \(P_1P_2\) as a directed line segment on the graph starting at \(P_1(3, 3)\) and ending at \(P_2(5, 5)\). This visual representation shows the direction and magnitude of the vector.

Step 4: Draw the Position Vector

The position vector of \(P_1P_2\) is the vector with an initial point at the origin \((0, 0)\) and terminal point at \(\langle 2, 2 \rangle\). Draw it on the graph from \((0, 0)\) to \((2, 2)\) effectively demonstrating the vector's components without reference to its starting position at \(P_1\).

Key Concepts

Vector Components

In vector mathematics, understanding vector components is crucial. A vector is defined by both its magnitude and direction.
To identify these characteristics, we break down a vector into its components.
These are the differences in the x and y values, respectively, between the points that the vector connects. For example, consider the vector from point \(P_1 = (3, 3)\) to point \(P_2 = (5, 5)\).
The components of this vector are found by subtracting the coordinates of \(P_1\) from \(P_2\).
The calculation is:\[(5 - 3, 5 - 3) = (2, 2)\] This tells us each step of the movement: - Move 2 units in the x-direction - Move 2 units in the y-direction The resulting vector can be expressed as \(\langle 2, 2 \rangle\).
Each component represents how far and in which direction you need to move to go from one point to the other.

Vector Graphing

Graphing vectors allows us to visualize direction and magnitude intuitively.
When graphing a vector like \(P_1P_2\), start by plotting both initial and terminal points on a coordinate plane.

• First place point \(P_1(3, 3)\) on the graph.
• Then mark point \(P_2(5, 5)\).
• Connect these points with a directed line segment.

This line segment is your vector \(P_1P_2\). It starts at \(P_1\) and ends at \(P_2\), displaying both the path and direction of the vector.
If you’re trying to visualize the vector’s magnitude, you’re effectively looking at the length and steepness of the line segment.
By using graphing, it becomes much easier to understand the physical interpretation of vectors.

Position Vector

The position vector is an important concept in vectors.
Unlike typical vectors, which can start anywhere, the position vector always starts from the origin \((0, 0)\).
It points to the vector's terminal point. This unique feature makes the position vector incredibly useful for graphing and calculating vectors without initially needing the actual starting point.For the vector \(P_1P_2\), the position vector is expressed as \(\langle 2, 2 \rangle\).
This representation simply means that if you start at the origin, you move 2 units right and 2 units up to get to the point \((2, 2)\). To illustrate, draw a vector from \((0, 0)\) to \((2, 2)\) on your chart.
This visually and symbolically represents the same length and direction as the original \(P_1P_2\) vector, but simplified by starting at the origin.