Vector Product Results in a vector

Dot product (Scalar product) Results in a scalar a · b = axbx+ayby+azbz Scalar

Vector Product Results in a vector =

Properties… a x b = - b x a a x a = 0 a x b = 0 if a and b are parallel. a x (b + c) = a x b + a x c a x (λb)= λ a x b

Examples… i x j = k, j x k = i, k x i = j j x i = - k, i x k = -j, k x j = i i x i = 0

c = a x b Vector product c a b c is perpendicular to a and b, in the direction according to the right-handed rule. a b c θ

Vector product – Direction: right-hand rule c = a x b a b c b a c θ

Vector product – right-hand rule c = a x b a b c c a θ b

Vector product – right-hand rule c = a x b c c b b θ a a

Vector Product-magnitude a = (a1, 0, 0) b = (b1, b2, 0) c a x θ b2 b y

Invariance of axb The direction of axb is decided according the right-hand rule. The magnitude of axb is decided by the magnitudes of a and b and the angle between a and b. a x b is invariant with respect to changes from one right-handed set of axes to another.

Application—Moment (torque) of a force

Moment of a force about a point M = | F |d F M = | F | |R |sinθ O R M = R x F θ d

Component of a vector a in an arbitrary direction s as --- Unit vector in the direction of s ax x

Example--Component of a Force F in an arbitrary direction s Fs --- Unit vector in the direction of s

Example--Component of a Moment M in an arbitrary direction s Fs --- Unit vector in the direction of s ---- Scalar Triple product

Scalar Triple Product Scalar

Volume of a parallelepiped = Volume of the parallelepiped. F E G bxc H a θ θ b D α C c A B

Moment of a force about an axis ---- Moment of F about axis AA’ --- Unit vector in the direction of s A’

Vector Triple Product Vector