Cross product of two vectors

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Definition. Cross product (vector product) of vector a by the vector b is the vector c, the length of which is numerically equal to the area of the parallelogram constructed on the vectors a and b, perpendicular to the plane of this vectors and the direction so that the smallest rotation from a to b around the vector c was carried out counter-clockwise when viewed from the terminal point of c (Fig. 1).

Fig. 1

Cross product formulas

Cross product of two vectors a = {a_x; a_y; a_z} and b = {b_x; b_y; b_z} in Cartesian coordinates is a vector whose value can be calculated using the following formulas:

a × b =	i	j	k	= i(a_yb_z - a_zb_y) - j(a_xb_z - a_zb_x) + k(a_xb_y - a_yb_x)
	a_x	a_y	a_z
	b_x	b_y	b_z

a × b = {a_yb_z - a_zb_y; a_zb_x - a_xb_z; a_xb_y - a_yb_x}

Cross product properties

Geometric interpretation.
The magnitude of the cross product of two vectors a and b is equal to the area of the parallelogram constructed on these vectors:
A_p = |a × b|
Geometric interpretation.
The area of the triangle constructed on the vectors a and a is equal to half the magnitude of the cross product of this vectors:

A_Δ = 1 |a × b|
2
Cross product of two non-zero vectors a and b is equal to zero if and only if the vectors are collinear.
The vector c that is equal to the cross product of non-zero vectors a and b, is perpendicular to these vectors.
c = a × b => c ┴ a and c ┴ b
a × b = -b × a
(k a) × b = a × (k b) = k (a × b)
(a + b) × c = a × c + b × c

Cross product examples

Example 1. Find the cross product of a = {1; 2; 3} and b = {2; 1; -2}.

Solution:

a × b =	i	j	k	=
	1	2	3
	2	1	-2

= i(2 · (-2) - 3 · 1) - j(1 · (-2) - 2 · 3) + k(1 · 1 - 2 · 2) =

= i(-4 - 3) - j(-2 - 6) + k(1 - 4) = -7i + 8j - 3k = {-7; 8; -3}

Example 2. Find the area of a triangle formed by vectors a = {-1; 2; -2} and b = {2; 1; -1}.

Solution: Calculate the cross product of these vectors:

a × b =	i	j	k	=
	-1	2	-2
	2	1	-1

= i(2 · (-1) - (-2) · 1) - j((-1) · (-1) - (-2) · 2) + k((-1) · 1 - 2 · 2) =

= i(-2 + 2) - j(1 + 4) + k(-1 - 4) = -5j - 5k = {0; -5; -5}

From the properties of the cross product:

A_Δ =	1	\|a × b\| =	1	√0² + 5² + 5² =	1	√25 + 25 =	1	√50 =	5√2
	2		2		2		2		2

Answer: A_Δ = 2.5√2.

Vectors Vectors Definition. Main information Component form of a vector with initial point and terminal point Length of a vector Direction cosines of a vector Equal vectors Orthogonal vectors Collinear vectors Coplanar vectors Angle between two vectors Vector projection Addition and subtraction of vectors Scalar-vector multiplication Dot product of two vectors Cross product of two vectors (vector product) Scalar triple product (mixed product) Linearly dependent and linearly independent vectors Decomposition of the vector in the basis

Online calculators with vectors

Tasks and exercises with vector 2D

Tasks and exercises with vector 3D

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