Problem 14 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

The vector \(\mathbf{v}\) in terms of \(\mathbf{i}\) and \(\mathbf{j}\) is \(\mathbf{v} = -8\mathbf{i} + 11\mathbf{j}\)

Step 1: Identifying Given Points

The given points are \(P_1 = (2, -5)\) and \(P_2 = (-6, 6)\). These represent the initial and terminal points of our vector respectively. Thus, \(x_1 = 2, y_1 = -5, x_2 = -6\) and \(y_2 = 6\).

Step 2: Calculating Each Coordinate

Plugging these values into our formula, we get the x-component of the vector to be \(x_2 - x_1 = -6 - 2 = -8\) and the y-component to be \(y_2 - y_1 = 6 - (-5) = 11\)

Step 3: Write the Vector in Terms of \(\mathbf{i}\) and \(\mathbf{j}\)

Now we have the x and y components of our vector, so we can write the vector \(\mathbf{v}\) in terms of \(\mathbf{i}\) and \(\mathbf{j}\) as \(\mathbf{v} = -8\mathbf{i} + 11\mathbf{j}\)

Key Concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is a powerful tool that allows us to analyze geometric shapes using algebra and the Cartesian coordinate system. It's a way to describe precisely the location of points, and the path of vectors, in a plane using pairs (or tuples) of numbers called coordinates.

In our exercise, the initial and terminal points of the vector are given as pairs of coordinates \(P_1 = (2, -5)\) and \(P_2 = (-6, 6)\). These coordinates represent positions on the x (horizontal) and y (vertical) axis. The change from the initial to the terminal point creates a vector. By finding the differences \(x_2 - x_1\) and \(y_2 - y_1\), we determine the vector’s components in each direction.

This concept is fundamental because it forms the basis of vector addition, subtraction, and even scalar multiplication. By understanding how to navigate the coordinate plane, students can visualize and solve complex problems in physics, engineering, and computer graphics.

Vector Notation

Vector notation is a concise way to represent vectors, which are quantities that have both magnitude and direction. The standard basis vectors in two dimensions are \(\mathbf{i}\) and \(\mathbf{j}\), corresponding to the unit vectors along the x-axis and y-axis, respectively.

When we write a vector like \(\mathbf{v} = -8\mathbf{i} + 11\mathbf{j}\), we're using vector notation to express that \(\mathbf{v}\) has an x-component of -8 (in the direction of \(\mathbf{i}\)) and a y-component of 11 (in the direction of \(\mathbf{j}\)).

Components of a Vector

A vector's components are its projections on the axes, which can be thought of as 'building blocks' for its construction. Finding these components, as we did in our exercise by subtracting the coordinates of the initial point from the terminal point, helps in understanding how vectors add together or decompose into simpler parts.

Vector Algebra

Vector algebra combines vectors using operations such as addition, subtraction, and scalar multiplication. It's an essential branch of mathematics that extends our understanding of algebra into multiple dimensions.

For example, vector addition involves summing the respective components of two vectors, while vector subtraction involves taking the difference of the components. Similarly, scalar multiplication involves multiplying each component of a vector by a scalar (a single number), which changes the vector's magnitude but not its direction, unless the scalar is negative which also will reverse the direction.

In the step-by-step solution provided for the exercise, we used vector subtraction to find the components of the vector connecting \(P_1\) to \(P_2\). These operations are not just theoretical; they have real-world applications in physics for describing forces, velocities, and many more physical phenomena.