# Section 11.1 ```Please do these problems on separate sheets of paper and staple them together.
Section 11.1
1. Plot the points (1, 0, 3), (0, −2, −1), (−1, 5, 2) on the same 3-dimensional coordinate
system.
2. Find the distance between the points (−10, 3, 4) and (−2, 1, 5).
3. Find the center and radius of a sphere with equation x2 +y 2 +z 2 +2x−6y −10z +34 = 0
4. Sketch the graph of 3x − 4y + 2z = 24. Make sure to label the axis intercepts.
5. Describe the graph of the equation xyz = 0.
6. Find an equation whose solutions are the set of all points that are the same distance
from the points (1, 2, −3) and (2, 3, 2).
7. Sketch the graph corresponding to the parametric equations x = 2 + t, y = 3 − t, z =
6 − 2t for −1 ≤ t ≤ 2
8. Find the length of the arc traced out by the parametric equations x = 2 cos(t), y =
2 sin(t), z = 3t for −π ≤ t ≤ π. (Hint: The arc length formula is
Z bp
(f 0 (t))2 + (g 0 (t))2 + (h0 (t))2 dt
L=
a
where f , g, and h are the functions of t corresponding to x, y, and z.)
Section 11.2
1. Draw each pair of vectors in a 2-dimensional coordinate plane, then do the indicated
operation both algebraically and using your drawing.
(a) u =&lt; 4, 2 &gt;, v =&lt; 1, −3 &gt;, calculate u + v
(b) u =&lt; 5, 0 &gt;, v =&lt; 4, −2 &gt;, calculate u − 2v
(c) u =&lt; 2, 6 &gt;, v =&lt; −3, 4 &gt;, calculate ||u|| and ||v||
2. Find a unit vector in the direction of the vector &lt; 3, 7, 5 &gt;.
3. A 200 pound weight rests on a frictionless 30 degree incline. Someone standing on
the incline above the weight’s position holds the weight in place with a rope and pulls
parallel to the incline. With how much force must they pull to keep the weight in
place?
1
4. The vectors u =&lt; 3, 2 &gt; and v =&lt; −1, 5 &gt; form a different basis for the plane than
the standard basis with i and j. This means that any vector in 2 dimensions may be
written as a linear combination of u and v. Write the vector &lt; 5, 3 &gt; as a combination
of these two vectors.
Section 11.3
1. For a =&lt; 3, 1, 0 &gt; and b =&lt; −2, 4, −1 &gt;, calculate the following:
(a) 2a − 4b
(b) a &middot; b
(c) ||a||2 − a &middot; a
(d) Then angle between a and b
(e) The projection of a onto b, projb a
(f) The projection of b onto a, proja b
(g) The equation of the plane going through (5, 0, −3) and perpendicular to b.
(h) An example of a vector that is perpendicular to both a and b.
2. Do Problem 1 again but for the vectors a =&lt; 1, −1, 4 &gt; and b =&lt; 2, 6, 1 &gt;.
3. For each expression, either calculate it or explain why it does not make sense. Use
p =&lt; −1, 3 &gt;and q =&lt; 5, 1 &gt;.
(a) (p &middot; q) + q
(b) (p + q) &middot; p
(c) 2 + p &middot; p
(d) (q &middot; p)||q||
(e) (q − 2p)2
2
```