Find the vector normal to the plane through the points

Question

Question 1.

(a) Find the vector normal to the plane through the points P(1,-1,0),Q(2,1,-1) and R(-1,1,2).

(b) Find the area of the triangle formed by the above three points.

Question 2.

Find the equation of the plane passing through the line of intersection of the two planes x+ y=2 and y-z=3 and which is perpendicular to the plane 2x+3y+4z=5.

Question 3.

Find the unit tangent vector, the principal normal vector, the curvature and the arc length of the curve over the interval 1≤ t ≤ 2 for the plane curve R = 2 In t i + (1/t +t) j. Find the equation of the circle of curvature when t=1.

Question 4.

Find the velocity and acceleration vectors, the speed ds /dt as well as the tangential and normal component of the acceleration for the motion described by R= t i +In t j for t >0.

Question 5.

For the curve R = (t- t³ /3, t², t+t³/3) find
(a) The unit tangent and normal vectors T(t) and N(t) at any point.
(b)Find the curvature k(t)