Final answer:
To find symmetric and parametric equations of the line where two planes intersect, calculate the cross product of the planes' normal vectors for direction and solve the equations for a specific point. Combine these to write the parametric equations and derive the symmetric equations.
Step-by-step explanation:
To find the symmetric and parametric equations of the line determined by the intersection of the planes 5x + y - 3z = 4 and x - y + 2z = 1, we need to solve the system of equations represented by these two plane equations. We can do this by finding a point of intersection and a direction vector for the line.
First, to find a direction vector, we can take the cross product of the normals to the two planes. The normal to the first plane is (5, 1, -3), and the normal to the second plane is (1, -1, 2). The cross product of these two vectors will give us a direction vector for the line.
To find a specific point on the line, we choose one variable to set as a parameter, typically 't', and solve for the other variables in terms of 't'. By substituting 't' back into one of the plane equations, we can find a corresponding point that lies on both planes, and thus on the intersection line.
Combining the point found and the direction vector, we can write the parametric equations of the line. The symmetric equations of the line are derived by solving each parametric equation for 't' and setting them equal