Mathematics 1

a × b = 0

a × b = −b × a - the vector product is anti-commutative

(a) × b = (a × b) = a × (b)- Scalar Multiplication Property

a × (b + c) = a × b + a × c –distributive property

(a + b) × c = a × c + b × c

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Vector Product in terms of vectors’ components

• For two vectors a and b defined by their components
• a( a1;a2;a3) and b( b1;b2;b3) the cross product is given by the formula:
• Easy to memorise using the fact that the cross product is really the determinant of a 3x3 matrix:

The Method of Cofactors to calculate:
where,

Vector Product in terms of vectors’ components -2D

• Two 2D vectors a and b can be defined by their components as a special case:
• a( a1;a2;0) and b( b1;b2;0), then the matrix for the cross product:

==-+= == (-

Then = =

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Vector Product in terms of vectors’ components

• The second method is easier but it will only work on 3x3 determinants
• .
• =a2b3+a3b1 +a1b2 -a1b3 -a3b2-a2b1 =(a2b3-a3b2)+ (a3b1- a1b3)+ (a1b2- a2b1)

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Vector Product -example

• If = (2;1;-1) and =(-3;4;1)compute each of the following:
• 1) 2)
• Solution:

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1)
2)

Geometrical interpretation of the Cross product magnitude

• We can give a geometrical interpretation to the magnitude of the vector product if we consider the parallelogram below.
• The area of this parallelogram is

the magnitude of the vector

product:

• Area = base × height = h =

sin() =

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h= v