The aim of this question is to become familiar with the **vector** representation. Two vectors are given in this question and their** product** needs to be found. After that, the visual representation of the origin is also made.

This question is based on the concepts of physics. **Vectors** are **quantities** that have **magnitude** as well as **direction**. There are two methods for vector multiplication: **dot product** and **cross product**. By performing the dot product, we obtain a scalar quantity that has only the magnitude but no direction, while the cross product results in a vector quantity. As we need a vector at the end of multiplication, therefore, we will perform a cross product.

## Expert Answer

We have **two vectors** $A$ and $B$:

\[ A(4, 0, -2) \]

\[ B(4, 2, 1) \]

These **vectors** can be represented with **end points** as follows:

\[ A(4, 0, -2) = A(x_1, y_1, z_1) \]

\[ B(4, 2, 1) = B(x_2, y_2, z_2) \]

In the above equations, $x, y,$ and $z$ show the **dimension** of the vectors in $x-axis, y-axis$, and $z-axis$, respectively. Hence, the required vector $\overrightarrow{AB}$ with the **end points** of vectors $A$ and $B$ can be written as follows:

\[ \overrightarrow {A B} = (x_2 – x_1) + (y_2 – y_1) + (z_2 – z_1) \]

\[ \overrightarrow {A B} = (4 – 4) + (2 – 0) + (1 + 2) \]

\[ \overrightarrow {A B} = 0 + 2 + 3 \]

\[ \overrightarrow {A B} (0, 2, 3) \]

## Numerical Results

A **vector** with directed **line segment** representation is as follows:

\[ \overrightarrow {A B} (0, 2, 3) \]

## Example:

Find the **directed line segment** $\overrightarrow {AB}$, given two points $A (3, 4, 1)$ and $B (0, -2, 6)$.

The **points** on the **graph** are given as:

\[ A (3, 4, 1) \]

\[ B (0, -2, 6) \]

If we represent the **coordinates** of the **cartesian plane** as:

\[ P (x, y, z) : \text{Where $P$ is any point on the graph and $x$, $y$, $z$ are its coordinate values} \]

We can represent the given points $A$ and $B$ as:

\[ A = (x_1, y_1, z_1) \]

\[ B = (x_2, y_2, z_2) \]

The **directed line segment** $\overrightarrow {AB}$ can be calculated by using the **distance formula:**

\[ \overrightarrow {AB} = (x_2\ -\ x_1, y_2\ -\ y_1, z_2\ -\ z_1) \]

Substituting the values from the given points:

\[ \overrightarrow {AB} = (0\ -\ 3, -2\ -\ 4, 6\ -\ 1) \]

\[ \overrightarrow {AB} = (-3, -6, 5) \]

The **directed line segmented** is calculated to be $\overrightarrow {AB} (-3, -6, 5)$.

*Images/ Mathematical drawings are created with Geogebra.*