TITLE: Seven Problems on Planes, Lines and Vectors in 3D # !b1 Problem1: Consider the unit sphere centered at the origin and the line with direction (2,1-3a,2-3a) passing through the point (0,a,a). When a=sqrt(2)/2 Find:
1. The points of intersection between the sphere and the line
2. The area of the triangle containning the pts in (1) and the north pole of the sphere.
Solution1 !b1 Problem2: Find three different vectors in 3D, u,v,w such that the cross product between them is not associative. Solution2 !b1 Problem3: Given a line with possition vector u and velocity v, find the coordinates of all the points at a fix distance R from the line. Deduce from there the equation of the cylinder. Solution3 !b1 Problem4: Show that the line !c0 (x-1)/3 = -y/2 = z+1 and the plane x + 2y + z = 1 are parallel. Solution4 !b1 Problem5: Find the formula for the angle between a line and a plane and use it to find the angle (in degrees) between the plane x+y-z = -1 and the line y = 1 - 3x on the xy-plane. Solution5 !b1 Problem6: Find the cross product and the inner product between the vectors (i+j) and (i-j+k). Solution6 !b1 Problem7: Is the distance between to parallel planes, Ax + By + Cz = D1 and Ax + By + Cz = D2 given by |D1-D2| ? If not, what is the correct formula. Find the distance between the planes, x+y+2z=2 and x+y+2z=4. Solution7 >