1. Vectors and Matrices
is just a unit vector. The
A is defined by
it is the unit vector lying along A and pointed like A (not like -A).
1A-1 Find the magnitude and direction (see the definition above) of the vectors
1A-2 For what value(s) of c will
ck be a unit vector?
1A-3 a) If
(1,3, -1) and Q
and dir A.
A vector A has magnitude 6 and direction (i
If its tail is at
(-2,0, I), where is its head?
1A-4 a) Let
P and Q be two points in space, and X the midpoint of the line segment PQ.
0 be an arbitrary fixed point; show that as vectors, OX
b) With the notation of part (a), assume that X divides the line segment PQ in
1. Derive an expression for OX in terms of OP and OQ.
1A-5 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
Is the second condition redundant?
1A-6 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).
1A-7 Let A
j be a plane vector; find in terms of a and b the vectors A' and A"
resulting from rotating A by 90'
(Hint: make A ttie diagonal of a rectangle with sides on the x and y-axes, and rotate the
c) Let i'
4j)/5. Show that
is a unit vector, and use the first part of the
exercise to find a vector
such that i',
forms a right-handed coordinate system.
1A-8 The direction (see definition above) of a space vector is in engineering practice often
given by its
To describe these, let A
ck be a space vector,
represented as an origin vector, and let a, p, and y be the three angles
that A makes
a) Show that dir A
cos cr i
(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,
are the direction cosines of a vector in space if
and only if they satisfy