Vector calculus

Vectors

a = (a1 , a2 , a3)

b = (b1 , b2 , b3)

Absolute value (length of the vector arrow) |a| = √(a12 + a22 + a32 )

Addition a + b = (a1+b1 ,a2+b2 , a3+b3) = b + a

Subtraction a - b(a1 -b1 ,a2- b2 , a3- b3)= - (b - a)

Subtraction b - a = (b1- a1 , b2- a2 , b3- a3) = - (a - b)

Multiplication by a constant k: ka = (ka1 , ka2 , ka3)

Skalar product (internal product) a · b = a1b1 +a2b2 + a3b3 = |a| |b| cos(a|b)

a perpendicular to ba|b = 90o ⇒cos( a|b) = 0 ⇒ Scalar Product = 0

Vector product (external product)  a x b = (a2b3-b2a3 , a3b1 - b3a1 , a1b2 - a2b1)

Vector product b x a= (b2a3- a2b3 , b3a1 - a3b1 , b1a2 - a1b2) = - a x b

Absolute value of vector product |a x b| = |a| |b| sin(a|b)

a parallel b a|b = 0o ⇒sin(a|b) = 0 ⇒ |vp| = 0

a x b and b x a are perpendicular to the plane common to a and b

a , b and a x b form a clockwise tripod

a, b, and b x a form a counter clockwise tripod

when going from a to b and then to the vector product.