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 b ⇒ a|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.