Problem 17 - University Calculus: Early Transcendentals

Question

Express each vector in the form \(\mathbf{w}=w_{1} \mathbf{i}+\) \(w_{2} \mathbf{j}+w_{3} \mathbf{k}\). \(\overrightarrow{P_{1} \vec{P}_{2}}\) if \(P_{1}\) is the point (5,7,-1) and \(P_{2}\) is the point (2,9,-2)

Step-by-Step Solution

Answer

The vector is \(-3\mathbf{i} + 2\mathbf{j} - 1\mathbf{k}\).

Step 1: Understand the Vector Notation

The vector \( \overrightarrow{P_{1}\vec{P}_{2}} \) is the directed line segment from point \( P_{1} \) to point \( P_{2} \). We can express it as \( \mathbf{w}=w_{1} \mathbf{i}+w_{2} \mathbf{j}+w_{3} \mathbf{k} \), where \(w_{1}, w_{2},\) and \(w_{3}\) are the components of the vector.

Step 2: Calculate the Components of the Vector

To find the components of vector \( \overrightarrow{P_{1}\vec{P}_{2}} \), subtract the coordinates of \(P_{1}\) from the coordinates of \(P_{2}\). This gives us: \[ \begin{align*} w_{1} &= 2 - 5 = -3,\ w_{2} &= 9 - 7 = 2,\ w_{3} &= -2 - (-1) = -1. \end{align*} \]

Step 3: Express the Vector in Component Form

Now that we have the components, we express \( \overrightarrow{P_{1}\vec{P}_{2}} \) in the form \( \mathbf{w}=w_{1} \mathbf{i}+w_{2} \mathbf{j}+w_{3} \mathbf{k} \). Thus, \( \overrightarrow{P_{1}\vec{P}_{2}} = -3\mathbf{i} + 2\mathbf{j} - 1\mathbf{k}\).

Key Concepts

Vector Components

Vector components are crucial in understanding and utilizing vectors in geometry and physics. Essentially, they break down a vector into its fundamental parts along the coordinate axes. If you have a vector expressed as \( \mathbf{w} = w_{1} \mathbf{i} + w_{2} \mathbf{j} + w_{3} \mathbf{k} \), each coefficient \((w_1, w_2, w_3)\) represents how much the vector moves along the x, y, and z axes, respectively.

For instance, in our example, the vector from point \(P_1\) to \(P_2\) is broken down into components, showing us how far and in what direction we move in three-dimensional space to travel from \(P_1\) to \(P_2\). Calculating these components involves subtracting the corresponding coordinates of the initial point from the terminal point.

Here's a step-by-step approach to find vector components:

• Choose two points in space, \(P_1(x_1,y_1,z_1)\) and \(P_2(x_2,y_2,z_2)\).
• Subtract the coordinates of \(P_1\) from \(P_2\): \(w_1 = x_2 - x_1\), \(w_2 = y_2 - y_1\), and \(w_3 = z_2 - z_1\).
• The result is a vector in component form: \(\mathbf{w} = w_1 \mathbf{i} + w_2 \mathbf{j} + w_3 \mathbf{k}\).

Breaking down vectors into components makes it easier to perform operations like vector addition, subtraction, and scalar multiplication.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, combines algebra and geometry to represent geometric shapes and solve geometrical problems using a coordinate system. It is particularly useful for dealing with vectors, which are represented as points in space defined by their coordinates.

The core idea is to transition from a purely conceptual or diagrammatic form of geometry to an algebraic form, allowing us to use mathematical formulas to find distance, midpoint, length, and areas. Vectors fit nicely in this framework because they can be represented as arrows in this space, originating from one point and reaching another.

By placing vectors in a coordinate system:

• You can easily calculate distances and directions, which are key in understanding motion and forces in physics.
• You can solve complex geometric problems by setting equations based on properties of lines, slopes, and distances.
• Understanding how components of a vector relate to the axes helps visualize how to break down forces or velocities along different paths.

For instance, in the example exercise, coordinate geometry allows seamless calculation of the vector from \(P_1(5, 7, -1)\) to \(P_2(2, 9, -2)\) through subtraction of coordinates to find the vector components.

Vector Subtraction

Vector subtraction is the process used to find a vector that represents the difference between two vectors or the displacement from one point to another. In simpler terms, when you subtract one point from another in a vector context, you are finding a vector that points from the first point to the second.

To perform vector subtraction:

• Align the two vectors' starting points and consider the movement required to go from the initial vector to the terminal vector.
• The difference in coordinates will give you the components of the resulting vector, as shown in \(w_1 = x_2 - x_1\), \(w_2 = y_2 - y_1\), \(w_3 = z_2 - z_1\).

In the exercise example, vector subtraction is used to obtain the vector \(\overrightarrow{P_{1}\vec{P}_{2}}\) by subtracting the coordinates of \(P_1\) from \(P_2\). This results in the vector \(-3\mathbf{i} + 2\mathbf{j} - 1\mathbf{k}\), highlighting the direct path and direction from \(P_1\) to \(P_2\).

Understanding vector subtraction is essential in many fields, including physics for calculating relative motion and engineering for determining stress vectors.