Problem 9 - Algebra and Trigonometry

Question

If \(\mathrm{v}\) is a vector with initial point \(\left(x_{1}, y_{1}\right)\) and terminal point \(\left(x_{2}, y_{2}\right),\) then which of the following is the position vector that equals \(\mathbf{v} ?\) (a) \(\left\langle x_{2}-x_{1}, y_{2}-y_{1}\right\rangle\) (b) \(\left\langle x_{1}-x_{2}, y_{1}-y_{2}\right\rangle\) (c) \(\left\langle\frac{x_{2}-x_{1}}{2}, \frac{y_{2}-y_{1}}{2}\right\rangle\) (d) \(\left\langle\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}\right\rangle\)

Step-by-Step Solution

Answer

(a) \langle x_2 - x_1, y_2 - y_1 \rangle

Step 1: Understand the question

Identify that we need to find the position vector \(\textbf{v}\) given the initial point \((x_1, y_1)\) and the terminal point \((x_2, y_2)\).

Step 2: Recall the position vector formula

The position vector from an initial point \((x_1, y_1)\) to a terminal point \((x_2, y_2)\) is calculated as \(\textbf{v} = \langle x_2 - x_1, y_2 - y_1 \rangle\).

Step 3: Apply the formula

Subtract the coordinates of the initial point from the coordinates of the terminal point. This gives us \(\textbf{v} = \langle x_2 - x_1, y_2 - y_1 \rangle\).

Step 4: Match with options

Compare the calculated position vector \(\textbf{v}\) to the provided options. The option that matches is (a) \langle x_2 - x_1, y_2 - y_1 \rangle\.

Key Concepts

Initial Point

In vector mathematics, the initial point is where the vector starts. It's often denoted as \((x_{1}, y_{1})\).
The initial point represents the location at the beginning of the vector in a two-dimensional plane.

When trying to find a vector, the initial point serves as a reference for starting the calculation.
It is crucial to distinguish this from the terminal point to compute the vector correctly

In our context, this is needed to apply the vector formula \((x_{2} - x_{1}, y_{2} - y_{1})\).
Always remember, identifying the correct initial point is the first step in accurate vector calculation.

Terminal Point

The terminal point defines where the vector ends. This point is often denoted as \((x_{2}, y_{2})\).
The terminal point shows the ending position of the vector in a plane.

It is essential to correctly identify the terminal point because it determines the direction and magnitude of the vector.
The difference between the terminal and initial points gives the components of the vector.
In the exercise, we need to subtract the coordinates of the initial point from the terminal point as per the formula used.

This subtraction provides us the change in both x and y directions, which forms the final position vector.

Vector Calculation

Vector calculation involves determining the vector components by using the initial and terminal points.
We use the formula:
\[ \textbf{v} = \langle x_{2} - x_{1}, y_{2} - y_{1}\rangle \]
This means subtracting the coordinates of the initial point from those of the terminal point.

Step-by-step, it works like this:

• Take the x-coordinate of the terminal point and subtract the x-coordinate of the initial point.
• Do the same for the y-coordinates.

This subtraction yields the components of the vector
For instance, if the initial point is (1, 2) and the terminal point is (4, 6), the position vector is:
\langle 4-1, 6-2 \rangle = \langle 3, 4 \rangle.
Combining these steps correctly provides the final solution.