Two completely different questions but I got stuck and wasn't sure on what to do for either. Any help?
Question 1: Three planes have equations:
Plane 1: x - 2y + z = 0
Plane 2: 3x - z = 4
Plane 3: x + y - z = k
a) Show that for all values of k, the planes do not intersect at a unique point. b) Find the value of k for which the intersection of the three planes is a line and find the vector equation of this line.
Question 2: A particle P of mass m is attached to one end of a light horizontal string. The other end of the string is attached to a fixed point. The magnitude of the tension in the string is given by 2mxk^2, where x is the extension in the spring at time t seconds and k > 0 is a constant. The particle experiences a resistance to motion of magnitude 3kmv, where v is the speed of the particle after t seconds.
a) Show that d^2 x/dt^2 + 3k(dx/dt) + 2k^2x = 0.
b) Given that t = 0, x = 4 and v = -3k:
i) Find x in terms of t and k
ii) State whether the damping is light, heavy or critical.
The only part i'm sure about for either is that I'm fairly certain the damping is light in Q2 b ii)
But i'm especially stuck on the whole of question 1 and the first part of Q2
Any help appreciated