Math 151 Week in Review
Monday Sept. 6, 2010
Instructor: Jenn Whitfield
Thanks to Amy Austin for contributing some problems.
All prolbems in this set are copywrited
Section 1.1
b) N 30◦ E (30 degrees east of North) at 30 mph.
6. A 500 pound bag of cement from two wires as shown
−→
in the figure below. Find the tensions (forces) T1
−→
and T2 in both wires and their magnitudes.
−→
The position vector of AB where A(x1 , y1 ) and
B(x2 , y2 ) is ~a = hx2 − x1 , y2 − y1 i.
Magnitude
of a vector ~a = ha1 , a2 i is
q
|~a| =
(a1 )2 + (a2 )2
Unit Vector: ~u =
1
~a
~a =
|~a|
|~a|
1. Given A(−5, 7) and B(−1, −2),
a) find the vector ~a with representation AB. Draw
AB and the corresponding position vector.
b) find the vector ~b with representation BA. Draw
BA and the corresponding position vector.
2. Given ~a = h−1, 2i and ~b = h4, 3i, compute the following and illustrate with a graph.
a) ~a + ~b
Section 1.2
Dot Product
~a • ~b = |~a||~b| cos θ, θ is angle between ~a & ~b and
0≤θ≤π
~a • ~b = a1 b1 + a2 b2 where ~a = ha1 , a2 i & ~b = hb1 , b2 i
~ where F
~ is force and D
~ is displacement
Work: F~ • D
1
b) − ~b
2
Two vectors are orthogonal if ~a • ~b = 0
Two vectors are parallel if ~a • ~b = ±|a||b|
c) ~a − ~b
Projection of ~b onto ~a
1
d) 2~a − ~b
2
~a • ~b
Scalar Proj: compa~b = |~b| cos θ =
|~a|
e) |~a|
3. Find a unit vector in the direction of ~a = 3~i − 2~j.
4. Given ~a = h6, −3i, ~b = h4, 8i, and ~c = h6, 2i,
(a) show, by means of a sketch, that there are
scalars s and t such that ~c = s~a + t~b.
(b) find the exact values of s and t.
5. A man walks due east on the deck of a ship at 4
mph. The ship is moving in the direction and speed
as indicated below. Find the direction and speed of
the man relative to the surface of the water.
a) Due north at a speed of 20 mph.
Vector Proj: proja~b =
~a • ~b
|~a|
!
(~a • ~b)(~a)
~a
=
|~a|
|~a|2
Orthogonal Complement for
~a = ha1 , a2 i is ~a⊥ = h−a2 , a1 i or ~a⊥ = ha2 , −a1 i
7. Find ~a · ~b given the following information:
a.) ~a = h−4, 7i and ~b = h2, 1i
b.) |~a| = 3, |~b| = 4, and the angle between ~a and ~b
is 30◦ .
8. Find the angle between the vectors h2, 0i and
h−1, 3i.
9. Find the value(s) of x so that ~a = hx, 2xi and ~b =
hx, −2i are orthogonal.
10. Find a unit vector that is orthogonal to ~a = −2~i + ~j
11. Find the scalar and vector projection of h1, 3i onto
h7, 5i.
12. Find the scalar and vector projection of h−2, 1i onto
h6, 1i.
13. Find the distance from the point (4, 1) to the line
y = 2x + 1.
Step 1 Pick two points on the
given line and find
position vector (~a).
Step 2:
Find the orthogonal
complement (~a⊥ )
Step 3:
Find another position
vector (~b) from the
given point to a point
from step 1.
Step 4:
Find the scalar projection
of ~b onto ~a⊥
14. A woman exerts a horizontal force of 25 pounds on a
crate as she pushes it up a ramp that is 10 feet long
and inclined at an angle of 20◦ above the horizontal.
Find the work done on the box.