In the following questions, do not evaluate surds (e. g. ? 2). leave them in your answer in that form (e. g. ? ? = 1 ± ? 2).
1. (a) find the vector and cartesian equations of the plane which contains the point a=(2,3,-1) and is perpendicular to the vector n =(1,2,2). what is the perpendicular distance of the plane from the origin?
?
(b) a plane has a normal = ? + 2? ? and is a perpendicular distance of 2 from the
origin. what are the vector and cartesian equations of the plane? what is the perpendicular distance of the plane from the point b = (3,1,-1)?
2. if ? ? 1 = 1 + 3? ? and ? ? 2 = 1 ? 2? ? express each of the following complex numbers in the form ? ? + where a and b are real numbers (standard form / cartesian form)
(a) (2? ? 1 + 1)
(b) (3? ? 2 ? 2)
(c) ? ? 1? ? 2
(d) , (complex conjugates of ? ? and ? ? )1212
(e) |? ? 1| , |? ? 2| (magnitudes of ? ? 1 and ? ? 2)
(f) 1z1
(g) ? ? 1? ? 2
3. if ? ? 1 =2+4? ? and ? ? 2 =4+? ? determine the complex number ? ? 3 =? ? 1? ? 2 .
sketch the complex plane to show the positions of all three of these complex numbers.
determine the modulus ? ? and (principal value) of the argument ? ? of each of these three complex).
4. solve ? ? 4 = ? 1?