**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.*