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Vectors are mathematical entities used to represent quantities with both magnitude and direction. In simpler terms, think of them as arrows that point from one place to another.
When working with points in space, vectors help describe the movement between those points. For instance, given two points A and B with coordinates \((2,2,2), (2,0,1)\), the vector from A to B is calculated by subtracting the coordinates of A from B.

This concept is crucial when proving that points form a specific shape, such as a rectangle. By analyzing vectors, we can determine distances (magnitudes) and angles between sides, confirming the geometric properties of the shape.

A rectangle is a four-sided polygon with opposite sides that are equal in length and each angle being 90 degrees. To prove that a set of points forms a rectangle using vectors:

• Calculate vectors between adjacent points
• Check the magnitudes of opposite sides
• Verify right angles between adjacent sides

In our example, the sides are represented by vectors \(\text{AB}, \text{BC}, \text{CD}, \text{DA}\). After computing these vectors, we confirmed their lengths and used the dot product to verify that each corner forms a right angle. This comprehensive approach using vectors validates that the given points form a rectangle.

The dot product is a way to multiply two vectors that produces a scalar. It is used to determine the angle between vectors. For two vectors \(\textbf{u} = (u_1, u_2, u_3)\) and \(\textbf{v} = (v_1, v_2, v_3)\), the dot product \(\textbf{u} \cdot \textbf{v}\) is given by:
\(\textbf{u} \cdot \textbf{v} = u_1v_1 + u_2v_2 + u_3v_3\)
In geometry, if the dot product is zero, the two vectors are perpendicular (form a right angle). For the rectangle example, we calculated the dot product between adjacent sides \(\textbf{AB} \cdot \textbf{BC}\), \(\textbf{BC} \cdot \textbf{CD}\), and so on.

Since all dot products were zero, this confirmed that each corner is a right angle, crucial for the rectangle's properties.

The magnitude of a vector is its length, which can be calculated using the Pythagorean theorem. For a vector \(\textbf{v} = (v_1, v_2, v_3)\), the magnitude \(||\textbf{v}||\) is:
\(||\textbf{v}|| = \sqrt{v_1^2 + v_2^2 + v_3^2}\)
Magnitudes help verify that opposite sides of the rectangle are equal. Let's consider vectors \(\textbf{AB} = (0, -2, -1)\) and \(\textbf{CD} = (0, 2, 1)\).
Their magnitudes are:
\(||\textbf{AB}|| = \sqrt{0^2 + (-2)^2 + (-1)^2} = \sqrt{5}\)
\(||\textbf{CD}|| = \sqrt{0^2 + 2^2 + 1^2} = \sqrt{5}\)
Since these magnitudes are equal, it shows that sides AB and CD are of equal length, which is essential for confirming that the figure is indeed a rectangle.