VectorsP1^
8
●^ Show^ that^ the^ angle^ between^ the^ three-dimensional^ vectors
a = a 1 i + a 2 j + a 3 k and b = b 1 i + b 2 j + b 3 k
is also given bycosθ= aa..bb
aabb
but that the scalar product a. b is now
a. b = a 1 b 1 + a 2 b 2 + a 3 b 3.Working in three dimensions
When working in two dimensions you found the angle between two lines by
using the scalar product. As you have just proved, this method can be extended
into three dimensions, and its use is shown in the following example.ExamPlE 8.13 The points P, Q and R are (1, 0, −1), (2, 4, 1) and (3, 5, 6). Find ∠QPR.SOlUTION
The angle between P→
Q and P→
R is given by θ in
→ →
cosθ= PQ→ PR→
PQ PR.
In thisP
→
PQQ ==
=
2
4
1
1
0
1
1
4
2
–
–^
| P
→
Q | = 1422 + + 22
= 21SimilarlyP
→
PRR ==
=
3
5
6
1
0
1
2
5
7
–
–^
| P
→
R | = 2522 + + 72
= 78ThereforeP
→
Q. P→
PQPRR =