91Ó°ÊÓ

Vectors are fundamental in analytical geometry and physics. They represent both magnitude and direction. You can think of them as arrows pointing from one point in space to another. In our exercise, vectors help describe the relative position of points in space.

**Forming Vectors**

When forming a vector between two points, you subtract the coordinates of the initial point from the final point. Given our exercise with points A, B, C, and D, the vectors are created by subtracting the coordinates:

• \( \vec{AB} = B - A = (-2, 4, 4) \)
• \( \vec{AC} = C - A = (-5, 8, 5) \)
• \( \vec{AD} = D - A = (-3, 3, 3) \)

These vectors represent directional movement from one point to another and are foundational in determining relationships between different points.

The determinant is a scalar value that can be calculated from a matrix. It's crucial in vectors to determine linear dependence and check coplanarity of points in space. In the context of understanding coplanarity, we use the determinant to assess if the vectors formed could lie on the same plane.

**Calculating the Determinant**
We arrange our vectors \( \vec{AB} \), \( \vec{AC} \), and \( \vec{AD} \) into a matrix to compute the determinant. This is expressed as:
\[\det\begin{pmatrix} -2 & 4 & 4 \ -5 & 8 & 5 \ -3 & 3 & 3 \end{pmatrix}\]

**Importance in Coplanarity Check**

By calculating this determinant, we get a single number. If this value is zero, it confirms that the vectors lie within the same plane, showing that the points are coplanar. If it's not zero, the points aren't coplanar. This makes the determinant a powerful tool in geometry.

Coplanarity refers to the concept where multiple points share the same geometric plane. In analytical geometry, this is crucial when analyzing the spatial alignment or configuration of points.

The main goal of the method in the provided exercise is to determine if the vectors formed by points A, B, C, and D all share a common plane.

**How to Check**

• Firstly, construct vectors between the points.
• Secondly, arrange these vectors in a matrix form.
• Lastly, compute the determinant of this matrix.

Based on our determinant calculation, a result of zero indicates that these vectors (and hence the original points) are indeed coplanar. In our exercise, since the determinant is 0, the points A, B, C, and D lie in the same plane. Understanding this concept is essential for spatial reasoning and plays an integral role in 3D modeling and many other applications.