[SOLVED] MA1023 Homework 7

1. Find parametric equations for the following lines.

1) The line through the point P (3,−4,−1) parallel to the vector u~ = h1,1,1i. 2) The line through P (1,−1,2) and Q(2,0,−1).

2. Give u~ = h2,−3,−1i, v~ = h1,4,−2i, find

• u~ v~;
• v~ ×u~;
• (u~ +v~)×(u~ −v~).

3. Give u~ = h4,−2,−4i, v~ = h1,2,−1i. Find a unit vector perpendicular to both u~ and v~.
4. Given three points P (1,−1,2), Q(2,0,−1) and R(0,2,1).

• Find the area of the triangle with vertices P , Q and R.
• Find a unit vector perpendicular to the plane passing through three points P (1,−1,2), Q(2,0,−1) and (0,2,1).

In exercise 5 and 6, find an equation for the given plane.

5. The plane through P₀(0,2,−1) with normal vector n~ = h3,−2,−1i.
6. The plane through P (1,1,−1), Q(2,0,2) and R(0,−2,1).

In exercise 7 and 8, ~r(t) is the position of a particle in space at time t.

• Find the particle’s velocity and acceleration vectors.
• Find the particle’s speed and direction of motion at the given value of t.

7. ~r(t) = h2cost,3sint,4ti and t = ;
8. ~r(t) = he^−t,2cos3t,2sin3ti and t = 0.