Please give detail explanation Given a ray with an origin at

Please give detail explanation~

Given a ray with an origin at (4, 5, 3) and a direction vector of (0.577, 0.577, 0.577); find its closest intersection point (if there is one) with a sphere whose centre is located at (3, 4, 5) with radius r = 7. What is the normal vector at the point of intersection? (10%) Given a ray with an origin at (1, 1, 1) and a direction vector of (-1, -1, -1); find its intersection point with the triangle with vertices (1, 0.0), (0. 1.0) and (0, 0, 1). Find the barycentric coordinates of this intersection point. (10%)

Solution

Eqn of line   (x-4) /.577 =t, (y-5)/.577=t, (z-3)/.577=t

eqn of sphere ( x-3)² +(y-4)² + (z-5)² = 49

substitute in sphere and solve for t, get t = +- 5.3775

Hence points of intersection of line are known.

normal vector will have dc 1/.577, again substitue for line and known value of t and solve.

2) eqn of line: (x-1)/-1 = (y-1)/-1 = (z-1)/-1 =t

find eqns of the sides of the triangle   (x-1)/1 =(y-0)/-1 =(z-0)/0

(x-0)/0= (y-1)/1 = (z-0)/-1

(x-0)/-1= (y-0)/0 = (z-1)1

find the line x-1=y-1=z-1   =t intersect the triangle along which side, the points of intersection must be same for x,y, z for both.. Find these rae not satisfied for all x,y,z at most for two (intersects a plane) not a line. pl check.

Best