Problem 18 - Algebra and Trigonometry

Question

In Exercises \(13-20\), let ve the vector from initial point \(P_{1}\) to terminal point \(P_{2}\). Write \(\mathbf{v}\) in terms of \(\mathbf{i}\) and \(\mathbf{j}\) $$P_{1}=(-1,6), P_{2}=(7,-5)$$

Step-by-Step Solution

Answer

The vector \(\mathbf{v}\) from \(P_1\) to \(P_2\) in terms of \(\mathbf{i}\) and \(\mathbf{j}\) is \(8\mathbf{i} - 11\mathbf{j}\)

Step 1 - Identify the given points

The given points are \(P_1=(-1,6)\) and \(P_2=(7,-5)\)

Step 2 - Compute the vector components

The \(x\) component of the vector \(\mathbf{v}\) from \(P_1\) to \(P_2\) is the difference \(x_2 - x_1\), which is \(7 - (-1) = 8\), and the \(y\) component is the difference \(y_2 - y_1\), which is \(-5 - 6 = -11\)

Step 3 - Write the vector in terms of components

The vector \(\mathbf{v}\) from \(P_1\) to \(P_2\) can now be written in terms of \(\mathbf{i}\) and \(\mathbf{j}\) as \(8\mathbf{i} - 11\mathbf{j}\). This means that the vector starts at \(P_1\) and goes 8 units in the \(x\) direction and 11 units in the \(-y\) direction to reach \(P_2\)

Key Concepts

Vector Components

When we talk about vector components, we're breaking down a vector into its basic parts that can be easily managed and understood. Imagine you are describing the movement from one point to another in a plane. The movement is not just one straightforward line; it's a combination of movements in the horizontal and vertical directions.

The idea is to simplify the vector by splitting it into two parts: an x-component and a y-component, which describe how far the vector moves in the x (horizontal) and y (vertical) directions, respectively. Given two points, say \(P_1(-1,6)\) and \(P_2(7,-5)\), the vector components can be found by:

• Subtracting the x-coordinate of \(P_1\) from \(P_2\): \(x_2 - x_1 = 7 - (-1) = 8\).
• Subtracting the y-coordinate of \(P_1\) from \(P_2\): \(y_2 - y_1 = -5 - 6 = -11\).

The vector from \(P_1\) to \(P_2\) can be represented as \(8\mathbf{i} - 11\mathbf{j}\), where 8 is the x-component and -11 is the y-component. This notation uses unit vectors to clearly show how a vector can be understood in terms of its components.

Unit Vectors

Unit vectors are vectors with a magnitude of 1. They serve as basic building blocks for constructing any vector by scaling and combining them. The most common unit vectors used in 2D coordinate geometry are \(\mathbf{i}\) and \(\mathbf{j}\).

\(\mathbf{i}\) represents a unit vector along the x-axis. It is used to indicate movement in a horizontal direction. Similarly, \(\mathbf{j}\) represents a unit vector along the y-axis, responsible for vertical movement. By using these unit vectors, any vector can be expressed as a combination of horizontal and vertical components.

For instance, the vector \(8\mathbf{i} - 11\mathbf{j}\) means moving 8 units in the x direction and -11 units in the y direction. It essentially breaks down the given vector into its core directions using these simple elements, enabling easy visualization and calculation.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a crucial aspect of understanding and utilizing vectors. It involves using a coordinate system to define and explain geometrical shapes and their properties. Within this system, points and vectors can be described using pairs of numbers which denote positions in space.

The exercise at hand uses coordinate geometry to dissect and simplify the movement from point \(P_1 = (-1, 6)\) to point \(P_2 = (7, -5)\) by using vector components. Each point in the plane has its own unique coordinates (x, y), allowing for precise calculations and visualizations.

Using this framework, the calculation of vector components becomes straightforward:

• The x-component is simply the difference between the x-coordinates of the two points: \(7 - (-1)\).
• The y-component is the difference between the y-coordinates: \(-5 - 6\).

Coordinate geometry provides a structured way to approach problems involving vectors, making complex spatial relationships easier to handle with basic arithmetic.