How to find a vector perpendicular to a plane?
Understand the Problem
The question is asking how to determine a vector that is perpendicular to a given plane. This involves understanding the geometric relationship between planes and vectors in three-dimensional space, typically using the normal vector of the plane.
Answer
The normal vector to the plane is $\vec{n} = \langle A, B, C \rangle$.
Answer for screen readers
The normal vector of the plane is given by $\vec{n} = \langle A, B, C \rangle$ from the plane equation $Ax + By + Cz + D = 0$.
Steps to Solve
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Identify the equation of the plane The equation of a plane in three-dimensional space can be given in the form $Ax + By + Cz + D = 0$. Here, the coefficients $A$, $B$, and $C$ represent the components of the normal vector to the plane.
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Extract the normal vector From the equation of the plane identified in the first step, the normal vector $\vec{n}$ can be extracted directly. The normal vector is represented as $\vec{n} = \langle A, B, C \rangle$.
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Write the normal vector explicitly If, for example, the equation of the plane is $2x - 3y + 4z + 5 = 0$, then the normal vector is: $$ \vec{n} = \langle 2, -3, 4 \rangle $$
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Confirm perpendicularity To confirm that the vector $\vec{n}$ is perpendicular to the plane, remember that two vectors are perpendicular if their dot product equals zero. For any vector $\vec{v} = \langle x_1, y_1, z_1 \rangle$ lying in the plane, you can check: $$ \vec{n} \cdot \vec{v} = A x_1 + B y_1 + C z_1 = 0 $$
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Conclusion The normal vector $\vec{n}$ is the vector that is perpendicular to the plane defined by the equation, and it provides the orientation and direction of the normal line to that plane.
The normal vector of the plane is given by $\vec{n} = \langle A, B, C \rangle$ from the plane equation $Ax + By + Cz + D = 0$.
More Information
Normal vectors are crucial in various applications, including calculating reflections, determining angles, and more in physics and engineering. Understanding the relationship between planes and normal vectors is foundational in vector calculus.
Tips
- Confusing parallel vectors with perpendicular vectors. Ensure to remember that two vectors are perpendicular only if their dot product equals zero.
- Not extracting the normal vector correctly from the plane equation. Always double-check that $A$, $B$, and $C$ correspond to the coefficients of the variables in the plane equation.
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