Determine whether the planes a1 x+b1 y+c1 z=d1 and a2 x+b2...

Determine whether the planes $a_{1} x+b_{1} y+c_{1} z=d_{1}$ and $a_{2} x+b_{2} y+c_{2} z=d_{2}$ are parallel, perpendicular, or neither. The planes are parallel if there exists a nonzero constant $k$ such that $a_{1}=k a_{2}, b_{1}=k b_{2}$, and $c_{1}=k c_{2},$ and are perpendicular if $a_{1} a_{2}+b_{1} b_{2}+c_{1} c_{2}=0$. $$ x+3 y+z=7, x-5 z=0 $$

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Key Concepts

Plane Equation in 3D

A plane in three-dimensional space can be expressed in the form ax + by + cz = d, where a, b, and c are coefficients that determine the orientation of the plane, and d is a constant that affects its position. This form is fundamental for understanding and analyzing the geometry of planes.

Normal Vector

The normal vector of a plane is a vector perpendicular to the plane, typically given by the coefficients (a, b, c) of the plane's equation. It is essential because it directly relates to the orientation of the plane and is used to determine relationships such as parallelism and perpendicularity between planes.

Parallelism Condition for Planes

Two planes are parallel if their normal vectors are scalar multiples of each other, which means there exists a nonzero constant k such that each corresponding coefficient of one plane is k times the coefficient of the other. This condition ensures that the planes have the same orientation in space.

Perpendicularity Condition for Planes

Two planes are perpendicular if the dot product of their normal vectors is zero. The dot product, computed as a1*a2 + b1*b2 + c1*c2, being zero indicates that the normal vectors — and hence the planes — are at right angles to each other. This concept is rooted in the properties of orthogonal vectors in Euclidean space.

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