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A normal vector is a vector that is perpendicular to a given surface or plane. In simpler terms, it points directly outwards from the plane and forms a 90-degree angle with any vector lying on the plane. This property makes the normal vector crucial for various applications, including computer graphics, physics simulations, and geometric problem-solving.
When dealing with a plane defined by three points, finding a normal vector helps in understanding the plane's orientation in space. This is often done by calculating the cross product of two vectors that reside on the plane (formed by the given points). For instance, in our example, by determining the vectors \( \overrightarrow{PQ} = \langle -3, 2, -5 \rangle \) and \( \overrightarrow{PR} = \langle -5, 0, 6 \rangle \) and finding their cross product, we derive a vector perpendicular to the plane.
Thus, the normal vector serves as a key mathematical tool for defining the spatial properties and orientation of planes.

The cross product is a mathematical operation used to find a vector that is perpendicular to two given vectors in three-dimensional space. It is denoted by \( \times \) and is an essential concept in vector calculus and physics for determining directions perpendicular to two interacting forces or directions.
In our exercise example, we calculated the cross product of \( \overrightarrow{PQ} \) and \( \overrightarrow{PR} \) to find a normal vector to the plane. The cross product is computed using a determinant formed from a 3x3 matrix composed of the unit vectors and the components of the two vectors involved:
\[ \overrightarrow{PQ} \times \overrightarrow{PR} = \left| \begin{array}{ccc} \mathbf{i} & \mathbf{j} & \mathbf{k} \ -3 & 2 & -5 \ -5 & 0 & 6 \end{array} \right| \]
The evaluated result gives us the vector \( \langle 12, 43, 10 \rangle \). This vector is perpendicular to both \( \overrightarrow{PQ} \) and \( \overrightarrow{PR} \), thus serving as the normal vector for the plane.
Understanding how to compute and apply the cross product is vital for solving geometric problems where perpendicular vectors are required.

The dot product is another fundamental operation involving two vectors. It results in a scalar value and is calculated by multiplying corresponding components of the vectors and then summing these products. The dot product has several uses, including determining the angle between vectors and checking perpendicularity.
In the context of confirming whether a vector is perpendicular to a plane, the dot product provides a simple verification method. If the dot product between a normal vector and any vector in the plane is zero, then these vectors are perpendicular by definition.
For example, in the final step of our problem, we find the dot product of the calculated normal vector \( \langle 12, 43, 10 \rangle \) with \( \overrightarrow{PQ} = \langle -3, 2, -5 \rangle \):
\[ \langle 12, 43, 10 \rangle \cdot \langle -3, 2, -5 \rangle = (12 \cdot -3) + (43 \cdot 2) + (10 \cdot -5) = -36 + 86 - 50 = 0 \]
Since the result is zero, this confirms the perpendicularity and correctness of our calculated normal vector. Understanding and using the dot product is essential for verifying vector directions in three-dimensional space.