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Mixed product of vectors

−→ −→ −→
− →
−→ −→
−→
− →
PROOF. According to the definition of a mixed product: a · b · c =( a × b ) · c = d · c .
( )
−→ −→ −→ y 1 z 1 z 1 x 1 x 1 y 1
;
Following formula (5.9), d = a × b = . Having multiplied the
;

x 2 y 2 z 2 x 2 x 2 y 2
− → −→
scalar product of vectors d and c , we will get:

−→ −→ −→ y 1 z 1 z 1 x 1 x 1 y 1 y 1 z 1 x 1 z 1
a · b · c = · x 3 + · y 3 + · z 3 = · x 3 − · y 3 +

y 2 z 2 z 2 x 2 x 2 y 2 y 2 z 2 x 2 z 2

x 1 y 1 z 1
x 1 y 1
+ · z 3 = x 2 y 2 z 2 . 2

x 2 y 2

x 3 y 3 z 3
Thus, the mixed product of three vectors equals to a determinant of the third order, consisting
from coordinates of these vectors, placed in rows.
Example 5.4. Calculate a volume of a pyramid with vertices in points:
A(1; 2; −1), B (3; 2; 1), C(0; 3; 1), D(4; −1; 2). ,

B

. . . . .
A C

D

Figure 5.7 – Volume of a pyramid

Solution. Calculate vectors: AB = (2; 0; 2), AC = (−1; 1; 2), AD = (3; −3; 3) (fig. 5.7). It
−→ −→ −→
is known that the volume of a pyramid, built on sides AB, AB, AB, equals to the 1/6 of a volume of a

1 −→ −→ −→
parallelepiped, built on the same ribs. So, V pyr = · AB · AC · AD . Calculating a mixed product of
6
vectors:
2 0 2 2 0 2

−→ −→ −→
AB · AC · AD= −1 1 2 = 3 · −1 1 2 = 3 · (2 + 2 − 2 + 4) = 18. Thus,

3 −3 3 1 −1 1
V pyr = 1 · 18 = 3(cub.un).
6
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