Determine if the following planes are parallel or intersecting. If they intersec

Question

Determine if the following planes are parallel or intersecting. If they intersect, find the parametric equations for the line of intersection: x

Explanation / Answer

A first degree equation in variables x, y and z represent a general plane in 3-D space. ax + by + cz + d = 0. Different formulas both vector and algebraic are available to write the equation of a plane based on given conditions. 1. Equation to a plane containing the point P (x0, y0, z0) and perpendicular to the direction = The vector form of the equation is n . r = n . r0. where r0 is the position vector 0, y0, z0>. The 3-D form of the equation is a(x - x0) + b(y - y0) + c(z - z0) = 0. Combining the constant terms, the equation to the plane can be written as ax + by + cz + d = 0. 2. Equation to a plane passing through a given point P(x0, y0, z0) and parallel to two vectors = 1, m1, n1> and = The vector form of the equation is = + s + t where is the position vector of P. The algebraic form of the equation is = 0 3. Equation of a plane containing two given points P(x1, y1, z1) and Q(x2, y2, z2) and parallel to a given vector = . The vector form of the equation is where and are position vectors of the points P and Q and t and s are parameters. The algebraic form of the equation is = 0 4. The equation of the plane passing through three given points P(x1, y1, z1), Q(x2, y2, z2) and R(x3, y3, z3). The vector form of the equation is = + s( - ) + t( - ) where , and are the position vectors of the points P, Q and R. The algebraic form of the equation is = 0 Coplanar Back to Top Three points define a plane and hence they are coplanar. If more than three points lie on the same plane, they are called coplanar points.

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