1. Vectors and Matrices
1A.
Vectors
Definition. A
direction
is just a unit vector. The
direction of
A is defined by
A
dir A
=

,
(A
#
0);
IAI
it is the unit vector lying along A and pointed like A (not like A).
1A1 Find the magnitude and direction (see the definition above) of the vectors
a) i+j+k
b) 2ij+2k
c) 3i6j2k
1A2 For what value(s) of c will
$
i

j
+
ck be a unit vector?
1A3 a) If
P
=
(1,3, 1) and Q
=
(0,1,
I),find A
=
PQ, (A(,
and dir A.
b)
A vector A has magnitude 6 and direction (i
+
2j

2 k)/3.
If its tail is at
(2,0, I), where is its head?
1A4 a) Let
P and Q be two points in space, and X the midpoint of the line segment PQ.
Let
0 be an arbitrary fixed point; show that as vectors, OX
=
$(OP
+
OQ)
.
b) With the notation of part (a), assume that X divides the line segment PQ in
the ratio
r
:
s, where
r
+
s
=
1. Derive an expression for OX in terms of OP and OQ.
1A5 What are the i j components of a plane vector A of length 3, if it makes an angle
of 30' with i and 60' with
j
.
Is the second condition redundant?
1A6 A small plane wishes to fly due north at 200 mph (as seen from the ground), in a
wind blowing from the northeast at 50 mph. Tell with what vector velocity in the air it
should travel (give the i j components).
1A7 Let A
=
a i
+
b
j be a plane vector; find in terms of a and b the vectors A' and A"
resulting from rotating A by 90'
a) clockwise
b) counterclockwise.
(Hint: make A ttie diagonal of a rectangle with sides on the x and yaxes, and rotate the
whole rectangle.)
c) Let i'
=
(3 i
+
4j)/5. Show that
i'
is a unit vector, and use the first part of the
exercise to find a vector
j
'
such that i',
j
'
forms a righthanded coordinate system.
1A8 The direction (see definition above) of a space vector is in engineering practice often
given by its
direction cosines.
To describe these, let A
=
a
i
+
b
j
+
ck be a space vector,
represented as an origin vector, and let a, p, and y be the three angles
(5
T)
that A makes
respectively with
i
,
j
,
and k
.
a) Show that dir A
=
cos cr i
+
cos Pj
+
cos
y
k
.
(The three coefficients are
called the direction cosines of A.)
b) Express the direction cosines of A in terms of a, b, c;
find the direction cosines
ofthevector i +2j +2k.
c) Prove that three numbers t, u,
v
are the direction cosines of a vector in space if
and only if they satisfy
t2
+
u2
+
v2
=
1.