Consider the planes 3x+5y+2z=1 and 3x+2z=0,find a unit vector v with positive first coordinate that is a parallel to both planes.

To find a unit vector v with a positive first coordinate that is parallel to both planes, we need to find the cross product of the normal vectors of the two planes.

Step 1: Find the normal vectors of the planes The normal vector of a plane is given by the coefficients of x, y, and z in the plane equation.

For the first plane, 3x + 5y + 2z = 1, the normal vector is (3, 5, 2).

For the second plane, 3x + 2z = 0 (which can be rewritten as 3x + 0y + 2z = 0 for clarity), the normal vector is (3, 0, 2).

Step 2: Find the cross product of the normal vectors The cross product of two vectors gives a vector that is perpendicular to both of the original vectors. In this case, it will give us a vector that is parallel to both planes.

The cross product of the normal vectors is:

(3, 5, 2) x (3, 0, 2) = (52 - 20, 23 - 23, 30 - 53) = (10, 0, -15)

Step 3: Normalize the vector To make the vector a unit vector, we need to divide each component by its magnitude.

The magnitude of the vector is given by:

||v|| = sqrt(10^2 + 0^2 + (-15)^2) = sqrt(100 + 225) = sqrt(325)

Dividing each component of the vector by its magnitude, we get:

v = (10/sqrt(325), 0, -15/sqrt(325))

Step 4: Check the first coordinate To ensure that the first coordinate is positive, we can multiply the vector by -1 if necessary.

In this case, the first coordinate is already positive, so we don't need to make any changes.

Therefore, the unit vector v with a positive first coordinate that is parallel to both planes is (10/sqrt(325), 0, -15/sqrt(325)).