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_{y}b_{z} - a_{z}b_{y}) - j(a_{x}b_{z} - a_{z}b_{x}) + k(a_{x}b_{y} - a_{y}b_{x}) |
a_{x} | a_{y} | a_{z} | ||
b_{x} | b_{y} | b_{z} |
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
- 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}.
= 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}
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}.
= 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}
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 | √0^{2} + 5^{2} + 5^{2} = | 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
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