Symmetric Equations for Line of Intersection of Two Planes
Consider the planes x + y + z = 1 and x - 2y + 3z = 1
a. Find the angle between the two planes
b. Find symmetric equations for the line of intersection of the two planes
To solve for the angle between two planes, start by recognizing that the angle is determined by the angle between their normal vectors. The normal vector of a plane can be derived from the coefficients of x, y, and z in the equation of the plane. For the first plane, the normal vector is based on the coefficients of x, y, and z, and similarly for the second plane. Once you have the two normal vectors, you can find the angle between them using the dot product formula. The dot product gives a relationship between the magnitudes of the vectors and the cosine of the angle, allowing you to isolate and solve for the angle.
To find the line of intersection of the two planes, recognize that the line lies in both planes, meaning it must satisfy both plane equations simultaneously. A standard method to find the symmetric equations for the line is to solve the system of equations formed by the two planes. This often involves solving for two of the variables in terms of the third, leading to parametric equations for the line. Converting the parametric form to symmetric equations gives a concise way of describing the line.
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