Analytic Geometry with Vectors

Usage: Type in all requested numbers rounded to two decimal places like 4.12 and check the buttons. Then submit your results. A completely objective Perl program will look at them.

1. Find the angle at the vertex P in the triangle with vertices at P(1,0,1), Q(2,-2,1), R(1,1,0).
(2 points) I'm using
the dot product the cross product the box product
(4 points) The angle is equal to

2. Find the equation of the line at which the planes 

             x+2y+z=4  and  x-3y+3z=1
intersect.
(2 points) I'm using
the dot product the cross product the box product
(2 points) A point on the intersection line is
( , , )
(3 points) A direction vector of the intersection line is
i + j + k

3. Find the distance from the point (3,2,-3) to the line given by the equation 

             x=1+t, y=2-t, z=3t.
(2 points) I'm using
the dot product the cross product the box product
(5 points) The distance is equal to

4. Find the area of the triangle with vertices at P(2,1,5), Q(4,6,2), R(1,0,1).

(2 points) I'm using
the dot product the cross product the box product
(4 points) The area is equal to

5. Find the arclength of the curve 

             r(t) = etcos(t)i+etsin(t)j+etk
between t=0 and t=1.
(2 points) I do the following
find an inscribed polygon integrate |r'(t)| from 0 to 1 ask my neighbor
(5 points) The arclength is equal to

6. The motion of a particle is given by 

             r(t) = ln(t)i+t2j+3tk.
Find the velocity v and the acceleration a at t=1. What are the tangential and normal components and the acceleration at t=1?
(3 points) The tangential component is equal to
(4 points) The normal component is equal to

7. Consider the curve given by 

             r(t) = cos(t)i+ln(cos(t))j+sin(t)k.
Find the curvature k, the unit tangent vector T, the principal normal vector N and the binormal vector B at t=0.
(2 points) The curvature is equal to
(2 points) T = i + j + k
(2 points) N = i + j + k
(2 points) B = i + j + k

8. The acceleration of a particle is given by 

             a(t) = ti+t2j+t3k.
Find r(t) for t=1 if 
             v(0) = i+j  and  r(0) = i.
(6 points) r(1) = i + j + k

return