Exercise Class Week 1 – Vectors - Answers
1. Write down the length r and the angle  measured anticlockwise from the x-axis of
each of the vectors a to e in the diagram. Not much to explain here – note that angles are
measured anticlockwise, because this is angle from x towards y, consistent with a righthand rule that gives z upwards out of the page, and with  = tan–1 y/x. Note that e is the
same vector as it would be if it were shifted to start – or finish – at the origin.
a: r = 4
= 0
b: r = 4
 = 90°
c: r = 4◊2
 = 45°
d: r = 2◊2
 = 135°
e: r = 2
 = 180°
OR
–2,
0
2. Write each vector in component form, v = (vx, vy). Again, e is the same vector as it
would be if it were shifted to start – or finish – at the origin.
a=
(4, 0)
b=
(0, 4)
c=
(4, 4)
d=
(–2, 2)
e=
(–2, 0)
3. Calculate the following scalar products, using u.v = uv cos where u and v are the
lengths and  is the angle between the two vectors
a.a =
44cos 0
= 16
a is like a (same direction, parallel)
a.b =
44cos 90°
=0
a is totally unlike b (a and b are orthogonal)
b.c =
44◊2cos 45°
= 16
b and c are neither orthogonal nor parallel
c.b =
4◊24cos 45°
= 16
and that doesn’t depend on the order taken
c.d =
4◊22◊2cos 90°
=0
c and d are orthogonal
b.d =
42◊2cos 45°
=8
d.e =
2◊22cos 45°
=4
a.e =
42cos 180°
= –8
a is like e (parallel)
4. Calculate the following scalar products, using u.v = uxvx + uyvy
a.a =
16 + 0
= 16
a.b =
0+0
=0
b.c =
0 + 16
= 16
c.b =
0 + 16
= 16
c.d =
–8 + 8
=0
b.d =
0+8
=8
d.e =
4+0
=4
a.e =
–8 + 0
= –8
Same answers as using uv cos 
Which method is easier?
5. Calculate the following vector products, using uv = uv sin
aa =
44sin 0°
=0
Area between is zero
ab =
44sin 90°
= 16
Area between is maximum
cd =
4◊22◊2sin 90°
= 16
Note that dc would be –16
6. Calculate the following vector products, using
uv = (uzvy – uyvz, uz vx – ux vz, ux vy – uy vx)
aa =
(0 – 0, 0 – 0, 0 – 0)
= (0, 0, 0)
ab =
(0 – 0, 0 – 0, 16 – 0) = (0, 0, 16)
cd =
(0 – 0, 0 – 0, 8 – –8) = (0, 0, 16)
Same answers as using uv sin 
But direction is given explicitly.
7. What are the following vector products (note: hats not easily available in Word)
ii =
0
Note that i = (1, 0, 0)
ij =
k
ik =
–j
As expected (Why?)
kj =
–i
Note how one can cycle the symbols
8. For each of the following relationships between two vectors both of length r = 1, what
can you say about the two vectors concerned?
Their dot product vanishes; they are . . . Orthogonal [OR at right angles,
perpendicular]
Their dot product vanishes; the angle between them is . . .
90°
Their dot product is 1; they are . . .
Parallel
Their dot product is 1; the angle between them is . . .
0°


3 1
3 1
,  and v   0,
,  .
9. Two vectors are u   3 ,
2 2

 4 4
Find their lengths u:
and v:
u2 = 3 + ¾ + ¼ = 4
2
v = 0 + 3/16 + 1/16 = ¼
so u = 2
so u = ½
Calculate their scalar product:
0 + 3/8 + 1/8 = ½
Hence find the angle between them:
uvcos = ½
so  = 60°
Write down the unit vectors uˆ : ½u = ½ (◊3, ½◊3, ½) = (½◊3, ¼◊3, ¼)
and vˆ :
2v = 2 (0, ¼◊3, ¼) = (0, ½◊3, ½) [These are the vector over its length.]
(Note: here the hats are essential, so I had to sort out how to get them in Word.)