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Multiplication of Vectors

Example 5.2. Three vertices of a triangle are given: A = (1; 2; 1), B =
(0; 4; 3), C = (−5; −1; 3). Find a cosine of angle φ near vertex A (fig. 5.2).,

−→
−→
Solution. Obtain vectors: AB = (−1; 2; 2), AC = (−6; −3; 2). Using formula (5.7), we will get:
−→ −→
AB · AC −1 · (−6) + 2 · (−3) + 2 · 2 4
cos φ = = √ √ =
2
2
2
2
2
−→ −→ (−1) + 2 + 2 · (−6) + (−3) + 2 2 21
AB · AC
B

φ

A . . . . . C
Figure 5.2 – Angle near vertex of triangle

5.2. Cross product of vectors

−→ −→
−→
Vectors ( a , b , c ) form the right triple of vectors, if we put these vectors to the same beginning
from the end of the third vector we will see the shortest turn from the first vector to the second
one against motion. On the contrary, a triple of vectors is called a left triple. Note, that a triple
−→ −→ −→
of base vectors ( i , j , k ) is the right one.

Definition 5.2. Vector c , that satisfies the below conditions:
−→
− →
−→
1. A module of vector c equals to the multiple of modules of vectors a

−→ −→ −→
−→
and b on a sine of an angle between these vectors: | c | = | a | · b ·
sin φ.
−→ −→
−→
− → −→ −→
−→
−→
2. Vector c is perpendicular to every vector a and b : c ⊥ a , c ⊥ b . a
−→
b
−→
− →
− →
3. Vectors a , b and c form the right triple of vectors (fig. 5.3) is called
− →
−→
cross product of vectors of a and b .
−→
−→
−→
The cross product is denoted by: c = a × b .

−→
−→
− →
−→
Note, that conditions 2, 3 take place, when | c | = | a | · b · sin φ ̸= 0, meaning c =

−→ −→ −→
−→ −→ −→ −→ −→
a × b ̸= 0. If | a | · b · sin φ = 0, then a × b = 0 (either at least one of vectors a or b
−→
−→
is a zero-vector or sin φ = 0, which takes place when vectors a or b are colinear).
The cross product also has its physical origin. Let’s consider such a physical task. M Sup-
−→ −→
pose, in point L of a solid body force F = LK is applied (fig. 5.4) and O is the point in space. It
−→
is known from mechanics that a moment of force related to a point is called a vector M (applied
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