Mathematical Investigations IV
Name:
Vectors
Getting To the Point
The Review
1.
Find the angle between the following vectors:
a.
v = <2, 5> and w = < –1, –4 >
a
>
3
b.
r = < a, 6> and s = < –2,
c.
p  3 iˆ  2kˆ and q  iˆ  ˆj  6kˆ
2.
For v = <2, 4, –1> and w = <-3, 3, 0>, find a unit vector in the v direction.
3.
Find scalars (a, b, c) such that v = ax + by + cz, where v = <3, –1, 5>,
x = <1, 2, 1>, y = <0, 4, –1>, and z = <–3, –2, 3>.
4.
Given p = < c, 3 >, q = < 7, 0 >, and r = < 2, –4 >, find c such that
a.
projqp = projqr
b.
p is perpendicular to r
c.
p and r are collinear
d.
c=r.q
Vectors. 10.1
Rev. F08
Mathematical Investigations IV
Name:
5.
Point A is located on the western bank of a river, and point C is located directly across
stream on the eastern bank. The river is 648.6 m wide and flows due south at 2.54 km/hr.
A boat wants to cross the river from point A to point C. Determine the bearing and
velocity that must be maintained by the boat in order to arrive at point C in 10 minutes.
6.
Find the magnitudes of T1 and T2.
T1
T2
40
55
-mgj = -30 kg  9.8 m/s2 j
7.
A boat heads in the direction N30E at a speed of 25 knots. The tide runs N80E at
10 knots. Find the resultant speed and direction of the ship. Support your results with
a picture.
8.
2
CB. Express the
3
following in terms of v and w.
C
D is a midpoint. CE =
w
D
E
v
A
B
AB =
DE =
BE =
BD =
Vectors. 10.2
Rev. F08
Mathematical Investigations IV
Name:
9.
Sketch projgh and projhg,
where g = < –4, 3 > and h = < 2, –5>
10.
Let v1 = <–3, 4>, v2 = <8, 15>, and v3 = <t , –8>.
v1  v2 =
a.
11.
b.
Determine the value(s) of t so that v3 and v2 are orthogonal.
c.
Determine the value(s) of t so that v1 and v3 are collinear.
Let v t = <2, 3> + t  <4, –5> for t = 1, 2, 3, ...
a.
Find 3 v1 – 2 v2 + v3 .
c.
Find the measure of the angle between v1 and v2 . Leave your answer rounded
to the nearest minute.
d.
Find
b.
Find | v2 |
50
v
j
j 1
12.
If v1 = <k – 5, –12> and v2 = <k2, k – 5>, determine all values of k so that
v1 and v2 are orthogonal.
Vectors. 10.3
Rev. F08
Mathematical Investigations IV
Name:
13.
Draw the projection of vector v onto
line l.
14.
Let u = <5, 7> and v = <–3, 10>. Find projuv.
15.
Find the scalars (a, b, c) such that w = a v1 + b v2 + c v3 , where w = <–4, 3, –1>,
v1 = <1, 2, 3>, v2 = <–2, 4, –1> and v3 = <0, 1, 5>.
16.
Let v = <–2, 2, 1> and u = <1, 4, –5>. Find each of the following.
a.
c.
w = v – 2u
projuv
b.
|v + u|
d.
the angle between v and w
Vectors. 10.4

Rev. F08
Mathematical Investigations IV
Name:
17.
Find the magnitude of the force on the
wires a and b.
18.
Classify each as a scalar or a vector.
19.
20.
a.
834
b.
16 v
d.
ut
e.
(vw) v
Draw v = <5, 6, 3>
c.
v–w
Draw w = <–5, –2, 4>
Let v = <2, a + 1> and w = <8, a – 7>. Determine the value of a so that the vectors are
parallel.
Vectors. 10.5
Rev. F08