This preview shows pages 1–3. Sign up to view the full content.
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.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document2
E. 18.02 EXERCISES
1A9
Prove using vector methods (without components) that the line segment joining the
midpoints of two sides of a triangle is parallel to the third side and half its length. (Call the
two sides
A
and
B.)
1A10
Prove using vector methods (without components) that the midpoints of the sides
of a space quadrilateral form a parallelogram.
1A11
Prove using vector methods (without components) that the diagonals of a parallel
ogram bisect each other. (One way: let X and Y be the midpoints of the two diagonals;
show X
=
Y
.)
1A12*
Label the four vertices of a parallelogram in counterclockwise order as OPQR.
This is the end of the preview. Sign up to
access the rest of the document.
 Spring '08
 Auroux
 Vectors, Matrices

Click to edit the document details