Problem 17 - 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 \(\mathbf{j}\) $$P_{1}=(-1,7), P_{2}=(-7,-7)$$

Step-by-Step Solution

Answer

The vector \(\mathbf{v}\) from \(P_{1}\) to \(P_{2}\) in terms of \(\mathbf{i}\) and \(\mathbf{j}\) is \(-6 \mathbf{i} -14 \mathbf{j}\).

Step 1: Identify the Coordinates of Points \(P_{1}\) and \(P_{2}\)

The given points are \(P_{1}=(-1,7)\) and \(P_{2}=(-7,-7)\). Therefore, the coordinates of both \(P_{1}\) and \(P_{2}\) are already known and can be directly used in the calculation.

Step 2: Apply the Vector Formula

Now, apply the vector formula \(\mathbf{v} = P_{2} - P_{1}\). This implies that the \(\mathbf{i}\) component (x-coordinate difference) is \(-7 - (-1) = -6\) and the \(\mathbf{j}\) component (y-coordinate difference) is \(-7 - 7 = -14\). Therefore the vector \(\mathbf{v}\) becomes \(-6 \mathbf{i} -14 \mathbf{j}\).

Step 3: Formulate the Final Vector

Combining the calculated components of the displacement vector, we obtain \(\mathbf{v} = -6 \mathbf{i} -14 \mathbf{j}\) which is the final form of the vector from \(P_{1}\) to \(P_{2}\) in terms of \(\mathbf{i}\) and \(\mathbf{j}\).

Key Concepts

Vector notation

Vectors are a fundamental concept in mathematics and physics, often represented in bold letters such as \( \mathbf{v} \), or with an arrow over a letter, like \( \vec{v} \). In two-dimensional space, vectors describe quantities that have both magnitude and direction, making them useful for tasks like representing forces or velocities.

One common way to express vectors is through unit vectors \( \mathbf{i} \) and \( \mathbf{j} \). These unit vectors are standard representations where \( \mathbf{i} \) denotes a unit vector in the horizontal direction (x-axis), and \( \mathbf{j} \) describes a unit vector in the vertical direction (y-axis). By using \( \mathbf{i} \) and \( \mathbf{j} \), any two-dimensional vector can be broken down into horizontal and vertical components.

In vector notation, the vector \( \mathbf{v} = a \mathbf{i} + b \mathbf{j} \) represents a vector with a magnitude of \( a \) units in the direction of \( \mathbf{i} \), and a magnitude of \( b \) units in the direction of \( \mathbf{j} \). This notation clarifies how vectors can be constructed, analyzed, and used in various calculations.

Vector components

Vector components refer to the parts of a vector that represent its impact along the x and y axes. When you have a vector, like the one defined by points \( P_1 \) and \( P_2 \), it is helpful to understand how far it stretches or compresses in each direction.

To determine the components of a vector, you perform simple subtraction using the coordinates of the initial and terminal points. For our exercise, with \( P_1 = (-1, 7) \) and \( P_2 = (-7, -7) \), the procedure is as follows:

• Calculate the x-component: subtract the x-coordinate of the initial point from the x-coordinate of the terminal point: \(-7 - (-1) = -6\)
• Calculate the y-component: subtract the y-coordinate of the initial point from the y-coordinate of the terminal point: \(-7 - 7 = -14\)

The calculated components \(-6\) and \(-14\) help in forming the vector as \( \mathbf{v} = -6 \mathbf{i} - 14 \mathbf{j} \). Here, \(-6 \mathbf{i}\) represents the horizontal displacement and \(-14 \mathbf{j}\) represents the vertical displacement.

Coordinate geometry

Coordinate geometry, or analytic geometry, is a powerful tool that integrates algebra and geometry using a coordinate system. It allows us to use algebraic methods to solve geometric problems and vice versa.

By plotting points like \( P_1 = (-1, 7) \) and \( P_2 = (-7, -7) \) on a Cartesian coordinate plane, you can visually assess distances and directions. Each point has an x-coordinate and a y-coordinate, representing its position along the x-axis and y-axis, respectively. This positioning is central to defining vectors and understanding their movement.

The beauty of coordinate geometry is in its simplicity and effectiveness in solving complex problems. For instance, to find the vector from one point to another, you apply the vector formula \( \mathbf{v} = P_2 - P_1 \). This subtraction reflects a key application of coordinate geometry, transforming geometric problem-solving into manageable algebraic computations that yield precise results. Each component of the vector derived through this method ties back to specific shifts from the initial to the terminal point.