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Find a vector $A$ with representation given by the directed line segment $AB$. Draw $AB$ and the equivalent representation starting from the origin $A(4, 0, -2), B(4, 2 ,1)$.

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:

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

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

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

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

Figure 1

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.

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