[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 particles velocity and acceleration vectors.
• Find the particles 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.