Problem 17 - Thomas Calculus

Question

Express each vector in the form \(\mathbf{v}=v_{1} \mathbf{i}+\) \(v_{2} \mathbf{j}+v_{3} \mathbf{k}.\) \(\overrightarrow{P_{1} P}_{2}\) 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 \( \mathbf{v} = -3\mathbf{i} + 2\mathbf{j} - \mathbf{k} \).

Step 1: Understand the Problem

We are asked to express the vector \( \overrightarrow{P_{1}P_{2}} \) in the form \( \mathbf{v} = v_1 \mathbf{i} + v_2 \mathbf{j} + v_3 \mathbf{k} \), where the points \( P_1 \) and \( P_2 \) have the coordinates \( (5, 7, -1) \) and \( (2, 9, -2) \) respectively.

Step 2: Calculate the Vector Components

The vector \( \overrightarrow{P_{1}P_{2}} \) can be found by subtracting the coordinates of \( P_1 \) from \( P_2 \). Calculate \( v_1 = 2 - 5 = -3 \), \( v_2 = 9 - 7 = 2 \), and \( v_3 = -2 - (-1) = -1 \).

Step 3: Write the Vector in Required Form

Now express the vector using the components found: \( \mathbf{v} = -3\mathbf{i} + 2\mathbf{j} - \mathbf{k} \). This is the form \( v_1 \mathbf{i} + v_2 \mathbf{j} + v_3 \mathbf{k} \).

Key Concepts

Vector Components

When dealing with vectors, understanding vector components is crucial. Vectors have both magnitude and direction, and in three-dimensional space, any vector can be broken down into three components. These components orient along the axes of the coordinate system, usually denoted by

• the x-axis, with the basis vector \( \mathbf{i} \)
• the y-axis, with the basis vector \( \mathbf{j} \)
• the z-axis, with the basis vector \( \mathbf{k} \)

For example, if you have a vector \( \overrightarrow{P_{1}P_{2}} \), its components \( (v_1, v_2, v_3) \) represent how far the vector extends along the x, y, and z axes respectively. In the problem, by subtracting the coordinates of point \( P_1 \) from \( P_2 \), we've identified the components as \( v_1 = -3 \), \( v_2 = 2 \), and \( v_3 = -1 \). Each component tells us the vector's reach in its respective direction, helping articulate vector motion precisely.

Coordinate Subtraction

Coordinate subtraction is a straightforward yet fundamental operation in vector mathematics. It is used to find the vector that connects two points in space. Given two points \( P_1 = (x_1, y_1, z_1) \) and \( P_2 = (x_2, y_2, z_2) \), you can calculate the vector \( \overrightarrow{P_{1}P_{2}} \) by subtracting each corresponding coordinate:

• The x-component is \( v_1 = x_2 - x_1 \)
• The y-component is \( v_2 = y_2 - y_1 \)
• The z-component is \( v_3 = z_2 - z_1 \)

This operation was applied in the exercise to find that \( v_1 = -3 \), \( v_2 = 2 \), and \( v_3 = -1 \). The subtraction method ensures your vector accurately represents the direction and distance from one point to another in the 3D space.

Vector Notation

In vector mathematics, expressing a vector in proper notation is pivotal for clarity and communication of ideas. Vector notation in three dimensions uses linear combinations of unit vectors \( \mathbf{i}, \mathbf{j} \), and \( \mathbf{k} \) to represent a vector.

• The unit vector \( \mathbf{i} \) points along the positive x-axis.
• The unit vector \( \mathbf{j} \) points along the positive y-axis.
• The unit vector \( \mathbf{k} \) points along the positive z-axis.

Thus, any vector \( \mathbf{v} \) in 3D space is expressed as \( \mathbf{v} = v_1 \mathbf{i} + v_2 \mathbf{j} + v_3 \mathbf{k} \), where \( v_1, v_2, \) and \( v_3 \) are its components along the x, y, and z directions. In our exercise, we concluded that our vector \( \overrightarrow{P_{1}P_{2}} \) can be neatly expressed as \( -3\mathbf{i} + 2\mathbf{j} - \mathbf{k} \). Using this notation highlights both the direction and the magnitude of each component.