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Mixed product of vectors

B

A . . . . C

Figure 5.5 – Area of triangle

−→
−→
−→
−→
Solution. Coordinates of vectors AB and AC are: AB = (1; 3; −1), AC = (2; 2; 1). It is known,

−→
−→
−→
that an area of parallelogram (fig. 5.5), built on vectors AB and AC: S paral = AB × AC . Obviously,
−→
−→
−→
the area of triangle △ ABC is: S ABC = 1/2 AB × AC . Let’s find a vector product of vectors AB and
−→
−→
AC:
− → −→ −→

i j k
−→ −→ 3 −1 − → −1 1 −→ 1 3 −→
AB × AC = 1 3 −1 = · i + · j + · k =

2 1 1 2 2 2
2 2 1

−→ − → − →
= 5 i + 3 j − 4 k = (5; 3; −4).
√ √
−→ −→ √ 5 2
2
2
2
It’s module is: AB × AC = 5 + 3 + (−4) = 5 2. Thus, S ABC = (sq.un.)
2
5.3. Mixed product of vectors
−→ −→
− →
Definition 5.3. A mixed (vector-scalar) product of three vectors a , b and c
is called a digit, which equals to the dot product of vector a on a cross product of
−→
− → −→ −→ −→ −→
vectors b and c . It is denoted by: a · b · c . ✓
−→ − → −→ −→ − → −→ − → −→
−→
So, according to the definition, a · b · c = a · ( b × c ) = ( a × b ) · c . Let’s formulate the
main properties of a mixed product.
1. On cyclic transposition of vectors, when the orientation of three vectors
− →
−→
−→
−→ −→
−→
−→ −→
−→
is kept, the mixed product does not change: a · b · c = b · c · a = c · a · b .
2. On changing places any two multipliers in a mixed product, meaning
changing the orientation of vectors, the mixed product will change its
− →
−→
− →
−→
−→
− →
−→ −→
−→
−→
−→ −→
sign on opposite: a · b · c =− b · a · c = − a · c · b = − c · b · a .
−→ − →
−→
− →
−→
−→
3. (λ · a ) · b · c = λ( a · b · c ) — a connecting property related to a scalar
multiplier.
To prove the above properties the definition of a mixed product and main properties of a
scalar and cross product should be used.
Let’s find out the geometrical essence of a mixed product of three vectors.
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