Problem 4 - Algebra and Trigonometry

Question

Find the vector v with initial point \(P\) and terminal point \(Q\) . $$ P(1,2,-1), Q(3,-1,2) $$

Step-by-Step Solution

Answer

The vector \(\mathbf{v}\) is \(\langle 2, -3, 3 \rangle\).

Step 1: Understanding the Problem

You're given two points, \(P(1, 2, -1)\) and \(Q(3, -1, 2)\), and need to find the vector \(\mathbf{v}\) from \(P\) to \(Q\). A vector is determined by the difference of the terminal point \(Q\) and initial point \(P\).

Step 2: Calculate the Difference

To find the components of vector \(\mathbf{v}\), subtract the coordinates of point \(P\) from \(Q\). This is done component-wise:\[ v_x = 3 - 1, \quad v_y = -1 - 2, \quad v_z = 2 - (-1) \]

Step 3: Perform the Subtractions

Now perform the actual subtractions:\[ v_x = 3 - 1 = 2 \]\[ v_y = -1 - 2 = -3 \]\[ v_z = 2 - (-1) = 2 + 1 = 3 \]

Step 4: Write the Vector

Combine the component results into vector \(\mathbf{v}\):\[ \mathbf{v} = \langle 2, -3, 3 \rangle \]

Key Concepts

Initial Point

In vector calculations, the initial point plays a crucial starting role. It marks the starting position of the vector and is essential for defining the direction from one point to another. Consider a situation where you've two points, such as in our exercise, the initial point is denoted as \( P(1, 2, -1) \). This point provides the first reference from which we'll subtract the coordinates of another point, known as the terminal point, to find the vector's direction and magnitude.
Understanding the initial point is the first step in vector construction. It sets the groundwork for all calculations that follow.
Once you've located this point in space, the next task is to determine where the vector ultimately ends, which leads us to the terminal point.

Terminal Point

The terminal point is where the vector ends after starting from the initial point. It completes the definition of the vector by giving it both direction and length. In the earlier exercise, the terminal point is given as \( Q(3, -1, 2) \).

• It indicates the vector's endpoint.
• It is essential for calculating the vector's components through coordinate subtraction.

The terminal point, just like the initial point, is usually given in three-dimensional space as shown with coordinates \( (x, y, z) \). Determining the vector involves finding the change in position from the initial to the terminal point. An understanding of these coordinate changes will allow us to extract the vector components.

Coordinate Subtraction

Coordinate subtraction is a vital mathematical operation used to find the vector's components. By subtracting the coordinates of the initial point \( P \) from the terminal point \( Q \), we determine how much movement is required in each dimension to go from \( P \) to \( Q \).

In our exercise:

• The X-component is given by \( v_x = 3 - 1 = 2 \).
• The Y-component is calculated as \( v_y = -1 - 2 = -3 \).
• The Z-component results in \( v_z = 2 - (-1) = 3 \).

This step involves simple arithmetic but highlights the importance of directional change. The signs of the components tell us whether the movement is increasing or decreasing along each axis. Coordinate subtraction paves the way for creating the actual vector components.

Vector Components

Vector components are the individual segments of a vector along the x, y, and z axes. Once you've subtracted coordinates for each dimension, these components represent the vector comprehensively.
In our given problem, the vector components are calculated as \( \mathbf{v} = \langle 2, -3, 3 \rangle \).

• \( 2 \) is the X-component, indicating two units movement in the positive x-direction.
• \( -3 \) is the Y-component, showing three units decrease or movement in the negative y-direction.
• \( 3 \) is the Z-component, reflecting three units increase in the z-direction.

When combined, these components offer a complete depiction of the vector's direction and magnitude in space. Understanding vector components is crucial as it forms the backbone of vector calculations and applications across various fields of study.