Problem 31 - Algebra and Trigonometry

Question

The vector \(\mathbf{v}\) has initial point \(P\) and terminal point \(Q .\) Find its position vector. That is, express \(\mathbf{v}\) in the form \(a \mathbf{i}+b \mathbf{j} .\) $$ P(-2,-1) ; \quad Q=(6,-2) $$

Step-by-Step Solution

Answer

\(\mathbf{v} = 8 \mathbf{i} - 1 \mathbf{j}\)

Step 1: Identify Coordinates

Note the given coordinates for points P and Q. Let P(-2, -1) be the initial point and Q(6, -2) be the terminal point.

Step 2: Find Differences in Coordinates

Calculate the difference between the corresponding coordinates of points Q and P. \[ \text{Difference in x-coordinates:} \ x_2 - x_1 = 6 - (-2) = 6 + 2 = 8 \ \text{Difference in y-coordinates:} \ y_2 - y_1 = -2 - (-1) = -2 + 1 = -1 \]

Step 3: Form the Position Vector

Combine the differences in coordinates to form the position vector. The position vector \(\mathbf{v}\) is represented as \(a \mathbf{i} + b \mathbf{j} \,\text{where}\, a = \text{difference in x-coordinates}\,\text{and}\, b = \text{difference in y-coordinates}\). Thus, \(\mathbf{v} = 8 \mathbf{i} - 1 \mathbf{j}\).

Key Concepts

Vector Notation

When working with vectors, it's essential to understand vector notation. Typically, vectors are expressed using bold letters or with an arrow over them. For instance, the vector can be denoted as \(\mathbf{v}\) or \(\vec{v}\). When dealing with vectors in a plane, we use coordinate points to specify their direction and magnitude.

For example, in the exercise given, the vector \(\mathbf{v}\) is defined by its initial point \(P(-2, -1)\) and terminal point \(Q(6, -2)\).

Vectors are often written in the form \(a \mathbf{i} + b \mathbf{j}\), where \(\mathbf{i}\) and \(\mathbf{j}\) are the unit vectors in the x and y directions, respectively.

Coordinate Difference

To find the position vector, we need to compute the coordinate differences between the initial and terminal points.

The x-coordinate difference: \[x_2 - x_1 = 6 - (-2) = 6 + 2 = 8\]
The y-coordinate difference: \[y_2 - y_1 = -2 - (-1) = -2 + 1 = -1\]

These differences tell us how far and in which direction we move from the initial point to the terminal point, forming the vector's components.

Understanding these differences lays the foundation for expressing the vector in its standard form. It's a simple yet crucial step.

Initial and Terminal Points

The initial and terminal points of a vector are crucial in defining its position and length. The initial point is where the vector starts, and the terminal point is where it ends.

In our example, the initial point \(P(-2, -1)\) serves as the starting location, while the terminal point \(Q(6, -2)\) gives the endpoint.

By calculating the differences in their coordinates, we form the position vector. This vector shows the direction and distance from the initial point to the terminal point: \(\mathbf{v} = 8 \mathbf{i} - 1 \mathbf{j}\).
Understanding these points helps in visualizing and comprehending how vectors operate within a coordinate system.