Problem 8 - Algebra and Trigonometry

Question

If the vector \(\mathbf{v}\) has initial point \(P,\) what is its terminal point? $$ \mathbf{v}=\langle 0,0,1\rangle, P(0,1,-1) $$

Step-by-Step Solution

Answer

The terminal point is (0, 1, 0).

Step 1: Understanding the Problem

We need to find the terminal point of the vector \( \mathbf{v} = \langle 0,0,1 \rangle \) when it starts from the initial point \( P(0,1,-1) \). A vector can be described by its change from an initial point to the terminal point.

Step 2: Identify the Components

The vector \( \mathbf{v} = \langle 0,0,1 \rangle \) represents the change in position. It doesn't affect the changes in the x- or y- components; the change is only in the z-component by 1 unit.

Step 3: Compute the Terminal Point

From the initial point \( P(0,1,-1) \), apply the vector \( \mathbf{v} \). Add each vector component to the corresponding initial point coordinate: \( x+0, y+0, z+1 \) which gives us the terminal point \( (0+0, 1+0, -1+1) \) resulting in \( (0, 1, 0) \).

Step 4: Write the Terminal Point

The terminal point of vector \( \mathbf{v} \) starting from \( P(0,1,-1) \) is \( (0, 1, 0) \).

Key Concepts

Terminal Point

In vector mathematics, the **terminal point** is the concluding position of the vector after accounting for the change specified by the vector components from a specific starting position called the initial point. You can imagine this as the final spot you reach if you travel along the vector from where you start. In simpler terms, if a vector indicates a direction and a distance, the terminal point is where you arrive after following this direction and distance. To determine the terminal point from an initial point, add each component of the vector to the corresponding coordinate of the initial point. For example, if the vector has components \(\langle a, b, c \rangle\) and originates from point \((x, y, z)\), the terminal point \((x', y', z')\) is calculated as:- \(x' = x + a\)- \(y' = y + b\)- \(z' = z + c\)This calculation applies the vector's directive changes in each spatial dimension to locate the endpoint.

Initial Point

The **initial point** essentially serves as the launch pad for our vector journey. It is the starting coordinate from which changes defined by the vector components are applied. In exercises, this point is crucial for accurately defining the motion of a vector in space.For example, let’s look at the initial point \(P(0,1,-1)\) and vector \(\mathbf{v} = \langle 0,0,1 \rangle\). The task is to determine where this will lead us when applied to the starting point:- Initial point coordinates are \(x = 0\), \(y = 1\), and \(z = -1\).
- To ascertain the terminal point, add the vector's components \(\langle 0,0,1 \rangle\) to the initial point. Thus, each component of the vector modifies the respective coordinate of the initial point to determine the ending point.

Vector Components

A vector in mathematics is fundamentally represented by its **components**, written in the format \(\langle a, b, c \rangle\) for three-dimensional space. These components describe how far the vector moves in each coordinate direction, namely X, Y, and Z.

• The first component \(a\) shifts the position along the x-axis.
• The second component \(b\) shifts it along the y-axis.
• The third component \(c\) shifts it along the z-axis.

When considering the vector \(\mathbf{v} = \langle 0,0,1 \rangle\), it implies no shift along the x or y directions, but a movement of one unit in the positive z direction. Understanding how these components work helps simplify the task of determining where the vector directs an object from its starting point to its terminal, or ending, point.