The(^) angle
(^) between
(^) two
(^) vectors
P1^
8
Further points concerning the scalar product
●●You will notice that the scalar product of two vectors is an ordinary
number. It has size but no direction and so is a scalar, rather than a
vector. It is for this reason that it is called the scalar product. There is
another way of multiplying vectors that gives a vector as the answer; it is
called the vector product. This is beyond the scope of this book.
●●The scalar product is calculated in the same way for three-dimensional
vectors. For example:
2
3
4
5
6
7
25 3 647 56
. = ×+× +× =.
In generala
a
ab
b
bab ab ab1
2
31
2
311 22 33
. =+ +
●●The scalar product of two vectors is commutative. It has the same value
whichever of them is on the left-hand side or right-hand side. Thus a. b = b. a,
as in the following example.2
36
7 263 733
. =× +× =
6
7
2
3
62 73 33
. = ×+× =.
●^ How^ would^ you^ prove^ this^ result?
The angle between two vectors
The angle θ between the vectors a = a 1 i + a 2 j and b = b 1 i + b 2 j in two dimensions
is given bycosθ=+
+× +
=
ab ab
aa bb11 22(^21222122)
aa..bb
aabb
where a. b is the scalar product of a and b. This result was proved by using the
cosine rule on page 271.