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 of the line of intersection of the two planes, we need to solve the system of equations represented by the two planes.

Step 1: Set up the system of equations The two plane equations are:

1. 3x + 5y + 2z = 1
2. 3x + 2z = 0

Step 2: Solve the system of equations We can solve the system of equations by substitution or elimination. In this case, we can subtract equation 2) from equation 1) to eliminate x:

(3x + 5y + 2z) - (3x + 2z) = 1 - 0 5y = 1 y = 1/5

Substitute y = 1/5 into equation 1): 3x + 5(1/5) + 2z = 1 3x + 1 + 2z = 1 3x + 2z = 0 x = -2z/3

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 of the line of intersection of the two planes is r = (t, 1/5, -2t/3).