Consider the planes 3x+5y+2z=1 and 3x+2z=0,find a vector form(with parameter t)for the line of intersectionof the two planes.

To find the vector form (with parameter t) for the line of intersection of the two planes, we need to find a common solution to both plane equations.

Step 1: Solve one of the equations for one of the variables Let's solve the second equation, 3x + 2z = 0, for x: x = -2z/3

Step 2: Substitute this into the other equation Substitute x = -2z/3 into the first equation, 3x + 5y + 2z = 1: 3(-2z/3) + 5y + 2z = 1 -2z + 5y + 2z = 1 5y = 1 y = 1/5

Step 3: Write the vector form of the line The vector form of a line in 3D space is given by r = a + tb, where r is the position vector of a point on the line, a is the position vector of a point on the line, t is the parameter, and b is the direction vector of the line.

In this case, a point on the line is (0, 1/5, 0) and the direction vector of the line is (1, 0, -2/3). So, the vector form of the line is:

r = (0, 1/5, 0) + t(1, 0, -2/3) = (t, 1/5, -2t/3)

Therefore, the vector form (with parameter t) for the line of intersection of the two planes is r = (t, 1/5, -2t/3).