Problem 76 - Algebra and Trigonometry

Question

Are based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. The vector \(\mathbf{v}\) has initial point \(P=(-1,2)\) and terminal point \(Q=(3,-4) .\) Write \(\mathbf{v}\) in the form \(a \mathbf{i}+b \mathbf{j} ;\) that is, find its position vector.

Step-by-Step Solution

Answer

The position vector is \[\mathbf{v} = 4\mathbf{i} - 6\mathbf{j}\].

Step 1: Identify the Initial and Terminal Points

The initial point of the vector \(\mathbf{v}\) is \(P=(-1,2)\), and the terminal point is \(Q=(3,-4)\).

Step 2: Use the Position Vector Formula

The position vector \(\mathbf{v}\) can be found using the formula: \[\mathbf{v} = (x_2 - x_1)\mathbf{i} + (y_2 - y_1)\mathbf{j}\]. Here, \(P=(x_1, y_1)=(-1,2)\) and \(Q=(x_2, y_2)=(3,-4)\).

Step 3: Calculate the Components

Substitute the given points into the formula. For the \(\mathbf{i}\)-component: \[a = x_2 - x_1 = 3 - (-1) = 3 + 1 = 4\]. For the \(\mathbf{j}\)-component: \[b = y_2 - y_1 = -4 - 2 = -6\].

Step 4: Write the Position Vector

Combine the components to write the vector in the form \[\mathbf{v} = a\mathbf{i} + b\mathbf{j}\]. Therefore, \[\mathbf{v} = 4\mathbf{i} - 6\mathbf{j}\].

Key Concepts

Vectors

Vectors are mathematical objects that have both magnitude (length) and direction. They are essential in physics and engineering because they allow us to describe quantities like force, velocity, and displacement. In a coordinate system, we can represent vectors graphically by arrows. The arrow's length represents the vector's magnitude, and the direction represents how it points in space. Vectors are different from scalars, which only have magnitude but no direction. For instance, 5 meters is a scalar, but a vector might describe moving 5 meters north.

Initial and Terminal Points

The initial and terminal points of a vector help to uniquely define its position and direction in space. The initial point, or tail, is where the vector starts, while the terminal point, or head, is where the vector ends. In our exercise, the vector \(\mathbf{v}\) starts at the initial point \(P = (-1, 2)\) and ends at the terminal point \(Q = (3, -4)\). By knowing these points, we can determine the vector's direction and calculate its components.

Vector Components

Vector components break a vector into parts that align with coordinate axes, making it easier to perform calculations. The components of a vector \(\mathbf{v}\) in two dimensions can be found using the differences in the x and y coordinates of its initial and terminal points. These components correspond to the changes in the x and y directions. For example, for the initial point \(P = (-1, 2)\) and the terminal point \(Q = (3, -4)\), the x-component change is \(3 - (-1) = 4\), and the y-component change is \(-4 - 2 = -6\). These are often written as the vector's components \(a\) and \(b\), so our vector can be expressed as \(4 \mathbf{i} - 6 \mathbf{j}\).

Position Vector Formula

The position vector formula transforms the differences in coordinates of initial and terminal points into vector notation. The formula is:
\[\mathbf{v} = (x_2 - x_1)\mathbf{i} + (y_2 - y_1)\mathbf{j}\].
The components \(a\) and \(b\) represent the x and y components of the vector respectively. By substituting \(P(x_1, y_1) = (-1, 2)\) and \(Q(x_2, y_2) = (3, -4)\), we calculate:
\[a = x_2 - x_1 = 3 - (-1) = 4\]
\[b = y_2 - y_1 = -4 - 2 = -6\].
This gives us the final vector's representation \[\mathbf{v} = 4 \mathbf{i} - 6 \mathbf{j}\]. The position vector formula simplifies finding vector components and helps accurately describe direction and magnitude.