Problem 17 - Thomas Calculus

Question

In Exercises \(17-22,\) 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 \( \overrightarrow{P_1 P_2} = -3 \mathbf{i} + 2 \mathbf{j} - \mathbf{k} \).

Step 1: Understanding Vector Components

First, let's identify that the vector \( \overrightarrow{P_1 P_2} \) is the directional vector from point \( P_1 \) to point \( P_2 \). It is calculated by subtracting the coordinates of \( P_1 = (5, 7, -1) \) from \( P_2 = (2, 9, -2) \).

Step 2: Calculate the Difference in Each Component

Subtract the x-coordinates, y-coordinates, and z-coordinates of \( P_1 \) from \( P_2 \): \[ v_1 = 2 - 5, \quad v_2 = 9 - 7, \quad v_3 = -2 - (-1) \].

Step 3: Simplify the Results

Now, calculate the simplified differences: \[ v_1 = 2 - 5 = -3, \quad v_2 = 9 - 7 = 2, \quad v_3 = -2 + 1 = -1 \].

Step 4: Express as a Vector

Finally, express the vector \( \overrightarrow{P_1 P_2} \) in terms of \( \mathbf{i}, \mathbf{j} \), and \( \mathbf{k} \): \[ \overrightarrow{P_1 P_2} = -3 \mathbf{i} + 2 \mathbf{j} - \mathbf{k} \].

Key Concepts

Vector Components

A vector in three-dimensional space is represented by its components along the x, y, and z axes. These components tell you how far you go in each direction. Each vector can be described by the change in its position from its starting point to its endpoint.
For example, to find the components of the vector \( \overrightarrow{P_1 P_2} \), we look at how the coordinates change from point \( P_1 \) to point \( P_2 \).

• The x-component \( v_1 \) shows the horizontal change: it is calculated as \( v_1 = x_2 - x_1 \).
• The y-component \( v_2 \) indicates the vertical change: it is determined by \( v_2 = y_2 - y_1 \).
• The z-component \( v_3 \) reflects movement along the depth: expressed as \( v_3 = z_2 - z_1 \).

These vector components are essential as they allow us to describe any vector in terms of its direction and magnitude. They are the building blocks used to express the vector in vector notation.

Vector Subtraction

Vector subtraction is an important operation that helps in finding the change from one vector to another. Imagine you are moving from one point in space to another. Vector subtraction can help us understand this movement.
To subtract vectors, you simply subtract each corresponding component. For the vector \( \overrightarrow{P_1 P_2} \), you perform these calculations:

• Subtract the x-coordinate of \( P_1 \) from \( P_2 \): \( v_1 = 2 - 5 = -3 \).
• Subtract the y-coordinate of \( P_1 \) from \( P_2 \): \( v_2 = 9 - 7 = 2 \).
• Subtract the z-coordinate of \( P_1 \) from \( P_2 \): \( v_3 = -2 - (-1) = -1 \).

This subtraction method is straightforward but crucial, as it determines the directional vector \( \overrightarrow{P_1 P_2} \) from the first point to the second. This directional vector points from the tail of the vector \( P_1 \) to the head at \( P_2 \). It offers a clear representation of movement in space.

Vector Notation

Vector notation is a concise way to express the direction and magnitude of a vector by using unit vectors. In a 3D coordinate system, these are often represented by \( \mathbf{i} \), \( \mathbf{j} \), and \( \mathbf{k} \) for the x, y, and z directions, respectively.
The vector \( \overrightarrow{P_1 P_2} \) can be written using these unit vectors as follows:

• The component along the x-axis, \( v_1 \), multiplies \( \mathbf{i} \).
• The component along the y-axis, \( v_2 \), multiplies \( \mathbf{j} \).
• The component along the z-axis, \( v_3 \), multiplies \( \mathbf{k} \).

So in our example, the vector is expressed as \( -3 \mathbf{i} + 2 \mathbf{j} - \mathbf{k} \).
This notation style is incredibly useful, as it succinctly provides all the information required to visualize and work with vectors in mathematical computations. It simplifies complex concepts by boiling them down into manageable parts, allowing you to manipulate vectors with ease.