Problem 20 - Advanced Engineering Mathematics

Question

Find the initial point of the vector \(P_{1} P_{2}=\langle-5,-1\rangle\) if its terminal point is \(\langle 4,7\rangle\).

Step-by-Step Solution

Answer

The initial point is \( P_1 = \langle 9, 8 \rangle \).

Step 1: Understand the problem

We are given a vector \( \overrightarrow{P_1P_2} \) with components \( \langle -5, -1 \rangle \) and the terminal point \( P_2 \) is \( \langle 4, 7 \rangle \). We need to find the initial point \( P_1 \).

Step 2: Recall the vector formula

The components of the vector \( \overrightarrow{P_1P_2} \) can be found by subtracting the coordinates of \( P_1 \) from \( P_2 \). Mathematically, this is expressed as \( P_2 - P_1 = \langle -5, -1 \rangle \).

Step 3: Write the equations for each component

For the x-component: \[ 4 - x_1 = -5 \]For the y-component:\[ 7 - y_1 = -1 \]

Step 4: Solve for the x-coordinate

Rearrange the x-component equation to solve for \( x_1 \):\[ 4 - x_1 = -5 \]\[ x_1 = 4 + 5 \]\[ x_1 = 9 \]

Step 5: Solve for the y-coordinate

Rearrange the y-component equation to solve for \( y_1 \):\[ 7 - y_1 = -1 \]\[ y_1 = 7 + 1 \]\[ y_1 = 8 \]

Step 6: Determine the initial point

The initial point \( P_1 \) is \( \langle 9, 8 \rangle \). This is the point from which the vector \( \overrightarrow{P_1P_2} = \langle -5, -1 \rangle \) starts.

Key Concepts

Vector Components

When dealing with vectors, understanding their components is crucial. Components are essentially the projection of the vector along the axes of the coordinate system. A vector in two dimensions like \( \langle a, b \rangle \) can describe changes along the x-axis and y-axis respectively.

• The x-component (or horizontal component) represents how much the vector moves along the x-axis.
• The y-component (or vertical component) describes the vector's movement along the y-axis.

For example, if a vector is \( \langle -5, -1 \rangle \), this means it goes 5 units in the negative direction of the x-axis and 1 unit in the negative direction of the y-axis.
Breaking vectors into components helps us analyze and compute with them more easily, especially when adding, subtracting, or finding the product of vectors.

Initial and Terminal Points of Vectors

In vector mathematics, the concept of initial and terminal points helps define the direction and magnitude of a vector. The initial point is where the vector begins, and the terminal point is where it ends.
The vector from an initial point \( P_1 \) to a terminal point \( P_2 \) can be represented as \( \overrightarrow{P_1P_2} \). If given the terminal point and vector components, as in our exercise with vector \( \overrightarrow{P_1P_2} = \langle -5, -1 \rangle \) and terminal point at \( \langle 4, 7 \rangle \), we can calculate the initial point by reversing the process:

• For the x-component: \( x_2 - x_1 = -5 \), we solve for \( x_1 \).
• For the y-component: \( y_2 - y_1 = -1 \), we solve for \( y_1 \).

Solving these will give the initial point \( \langle 9, 8 \rangle \), allowing us to fully describe the vector's journey from start to finish.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, involves using algebra to study geometric figures. This approach uses coordinate systems to simplify the understanding of shapes and vectors in a plane.
By placing points, lines, and vectors on a coordinate plane, we can accurately describe their positions and relationships mathematically. For vectors, this means:

• Identifying positions with ordered pairs \( (x, y) \).
• Using these pairs to perform vector operations like addition, subtraction, and scalar multiplication.

In our exercise, we used coordinate geometry principles to find the initial point of a vector by employing subtraction. We calculated the differences in coordinate values to ascertain the components of the vector and reversed this process to find missing points.
Thus, coordinate geometry provides a powerful method to analyze vectors and solve geometric problems systematically.