Let a 213 b 101 c 013 What is the equation of a the line

Let a = (2,1,-3), b = (1,0,1), c = (0,-1,3). What is the equation of:

(a) the line through a parallel to b;

(b) the line through b and c;

(d) the plane through c parallel to a and b;

(e) the sphere with center a and radius 2?

a) Equtation of line pasing through a= ( 2, 1, -3) and parallel to vector < 1, 0,1>

r = < 2 ,1 ,3> +t< 1, 0,1>

= ( 2 +t , 1 , 3 +t )

b) b= ( 1, 0, 1) and c= ( 0, -1 ,3)

direction vector = (0, -1 , 3) - ( 1, 0 ,1) = ( -1 , -1 , 4).

The parametric equation of the line is then

(x,y,z) = ( 1 , 0 , 1) +t ( -1 , -1 , 4) = ( 1-t , -t , 1+4t)

e) Equation of sphere : (x- a)^2 +y(-b)^2 + ( z-c)62 = r^2

Centre ( a, b ,c) ---> ( 2, 1, -3) and r= 2

So, equation of sphere: (x-2)^2 + (y -1)^2 + (z+3)^2 = 4