Vectors & Matrices
Formative assignment week 2
1. Let O be a fixed origin in 3-space, let P and Q be any points in this
space, with position vectors p and q respectively, and let R be the
point such that the figure OP QR is a parallelogram. For each of the
following bound vectors, determine an expression in terms of p and q
for the free vector represented by that bound vector:
−→
(i) OP ;
−→
(ii) QO;
−→
(iii) OR;
−→
(iv) P Q;
−→
(v) QR;
−→
(vi) RP .
In each case explain your reasoning. Draw a diagram to help you think.
2. Let A, B, C, D be any four points in 3-space, and let P , Q, R, S be
the respective mid-points of the line segments AB, BC, CD, DA.
(a) Determine the position vectors p, q, r, s of P , Q, R, S in terms
of the position vectors a, b, c, d of A, B, C, D.
(b) Using these position vectors p, q, r, s, determine expressions in
−→
a, b, c, d for the vector represented by P Q and for the vector
−→
represented by SR.
(c) Conclude that P QRS must be a parallelogram (even though the
figure ABCD we started with certainly need not be one).




1
2
3. Let a =  −1  and b =  −3 . Determine:
−4
2
(a) 3a − 5b;
(b) |a + 2b|;
(c) the vector of length 3 in the opposite direction to b;
(d) a vector equation for the line through the points A and B with
position vectors a and b respectively.
4. Each of the following equations or conditions defines a geometric object
in 3-space. Describe in words what this object is.
(a) r = λk for some λ ∈ R;
(b) |r| = 1;
(c) x = y = z;
(d) x2 = y 2 = z 2 ;
(e) x ∈ Z, y ∈ Z, z ∈ Z;
(f) x ≥ 0, y ≥ 0, z ≥ 0, x + y + z ≤ 1.
[Note: As usual, the object defined 
by each
 condition is the set of all
x
points R whose position vector r =  y  satisfies the condition.]
z
5. Fill in the argument of Section 1.6 of the notes in detail.