Calculus

Vectors

1. Let ~a =< 2, 3 > and ~b = −î + 5ĵ (a) Graphically show ~a +~b (b) Compute ~a + 3~b

(c) Graphically show ~a−2~b (d) Find |~a|

2. Let ~v =< 2, 3,−5 > and ~w = −î + 5ĵ (a) Compute 2~v −3~w (b) Find the magnitude of ~w (c) Find a unit vector in the direction of ~v (d) Find a vector of length 1 in the opposite direction of ~v (e) Find a vector twice as long as ~w in the opposite direction (f) Find a vector half as long as ~v

3. Is < 1, 1, 1 > a unit vector? Why or why not?

4. Graph the following planes: (a) x−2y + 4z = 16 (b) z = 2 (c) y + 4z = 8 (d) y = −2