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HW 7, MA 1023 SOLVED

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HW 7, MA 1023
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
(1) ~u · ~v;
(2) ~v × ~u;
(3) (~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).
1) Find the area of the triangle with vertices P , Q and R.
2) 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.
1) Find the particle’s velocity and acceleration vectors.
2) Find the particle’s speed and direction of motion at the given value of t.
7. ~r(t) = h2cos t,3 sint,4ti and t =
π
2
;
8. ~r(t) = he
−t
,2cos 3t,2 sin 3ti and t = 0.

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