
Cross product of two vectorsPage Navigation:
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 counterclockwise when viewed from the terminal point of c (Fig. 1).
Cross product formulasCross 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 = {a_{y}b_{z}  a_{z}b_{y}; a_{z}b_{x}  a_{x}b_{z}; a_{x}b_{y}  a_{y}b_{x}}
Cross product properties
Cross product examplesExample 1. Find the cross product of a = {1; 2; 3} and b = {2; 1; 2}.
Solution:
= 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:
= 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:
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
Scalarvector 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
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