Problem 5 - Algebra and Trigonometry

Question

Find the vector v with initial point \(P\) and terminal point \(Q\) . \(P(6,-1,0), Q(0,-3,0)\)

Step-by-Step Solution

Answer

The vector \( \mathbf{v} \) is \((-6, -2, 0)\).

Step 1: Understanding the Problem

To find the vector \( \mathbf{v} \) with initial point \( P(6,-1,0) \) and terminal point \( Q(0,-3,0) \), we need to subtract the coordinates of point \( P \) from the coordinates of point \( Q \). This step is crucial as a vector \( \mathbf{v} \) from \( P \) to \( Q \) is represented by \( \mathbf{v} = PQ = (Q_x - P_x, Q_y - P_y, Q_z - P_z) \).

Step 2: Finding the x-component

Subtract the x-coordinate of point \( P \) from the x-coordinate of point \( Q \):\[ Q_x - P_x = 0 - 6 = -6 \]This gives the x-component of the vector \( \mathbf{v} \).

Step 3: Finding the y-component

Subtract the y-coordinate of point \( P \) from the y-coordinate of point \( Q \):\[ Q_y - P_y = -3 - (-1) = -3 + 1 = -2 \]This gives the y-component of the vector \( \mathbf{v} \).

Step 4: Finding the z-component

Subtract the z-coordinate of point \( P \) from the z-coordinate of point \( Q \):\[ Q_z - P_z = 0 - 0 = 0 \]This gives the z-component of the vector \( \mathbf{v} \).

Step 5: Combine Components to Form the Vector

Now that we have all components, combine them to form the vector \( \mathbf{v} \):\[ \mathbf{v} = (-6, -2, 0) \]This is the vector from point \( P(6,-1,0) \) to point \( Q(0,-3,0) \).

Key Concepts

Coordinates Subtraction

In vector calculation, subtracting coordinates is a fundamental process used to determine the components of a vector, particularly when given two points. Let's break this down step-by-step to understand this concept more clearly.

The coordinates of a vector are determined by subtracting the coordinates of the initial point from those of the terminal point. Given an initial point \(P(x_1, y_1, z_1)\) and a terminal point \(Q(x_2, y_2, z_2)\), the vector \(\mathbf{v}\) from \(P\) to \(Q\) is represented as:

• \(x\)-component: \(x_2 - x_1\)
• \(y\)-component: \(y_2 - y_1\)
• \(z\)-component: \(z_2 - z_1\)

This process essentially takes the difference between corresponding coordinates of the terminal and initial points.

For example, if the initial point is \(P(6, -1, 0)\) and the terminal point is \(Q(0, -3, 0)\), the vector \(\mathbf{v}\) is determined by subtracting the coordinates of \(P\) from \(Q\), yielding \((-6, -2, 0)\).

Initial and Terminal Points

Initial and terminal points are crucial in the study of vectors, as they define the direction and magnitude of the vector. By understanding these points, you can determine the path or flow that a vector represents.

The initial point \(P\) is the starting point of the vector. It represents where the vector begins. In our exercise, the initial point is \(P(6, -1, 0)\).
The terminal point \(Q\) is the endpoint of the vector. It indicates where the vector points to or where it ends. For our example, this is point \(Q(0, -3, 0)\).

In computation, clearly identifying these points is important as it sets up the problem for coordinate subtraction. Knowing which point is initial and which is terminal will determine in which direction the calculations go, affecting the resulting vector's directionality.

Vector Components

Vector components are the individual parts that combine to form a complete vector. They provide detailed information about the vector's orientation in space.

A vector in three-dimensional space is represented in terms of its components along the x, y, and z axes. For a vector \(\mathbf{v} = (a, b, c)\):

• The x-component \(a\) indicates how far the vector moves horizontally.
• The y-component \(b\) shows the vertical movement.
• The z-component \(c\) represents movement along the third axis, which is perpendicular to both x and y.

Each component can be found by carrying out the coordinate subtraction process we previously discussed.

In our example, the vector \(\mathbf{v} = (-6, -2, 0)\), derived from the initial point \(P(6, -1, 0)\) and the terminal point \(Q(0, -3, 0)\), has components that describe movement in three dimensions, making it possible to visualize or utilize the vector in space-based calculations.