1. Vector Product
Results in a vector

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

3. Vector Product
Results in a vector =

4. 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

5. 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

6. 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

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

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

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

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

11. 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.

12. Application—Moment (torque) of a force

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

14. Component of a vector a in an arbitrary direction sas
--- Unit vector in the direction of s ax x

15. Example--Component of a Force F in an arbitrary direction sFs
--- Unit vector in the direction of s

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

17. Scalar Triple Product
Scalar

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

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

20. Vector Triple Product
Vector