Vector, Forces

The Dot Product of a vector represents the scalar quantity of the combined vector, theoritically, only when 2 vectors of the same magnitude and opposite direction will cancel each other just like in forces, but according to the
A = ai + bj
B = ci + dj
a*b + c*d = |A||B|cosC

when angle C becomes 90 degrees, the Dot Product will become zero, what does this actually mean? the magnitude of the combined vector becomes zero? or the two vectors does not influence each other?
If the two vectors does not influence each other, why vectors without an angle of 90 degrees can be calculated to get a magnitude since they can be seperated into the x and y axis also.?

From my point of view, I think that the flaw of the vector rule lies at the interchanging of sides of the triangle when the angle becomes 90 degrees, cosine is no longer the case at this particular point, instead tangent should be used to obtain the actual angle and magnitude. Or maybe, tangent is the ultimate solution for the angle and the pathagoreom theorem is the ultimate solution for the magnitude by having to seperate everything into x,y,z axis beforehand.

After reviewed the material from MIT OCW, multivariable calculus lecture 1, The application of the vector is just to obtain the angle of 2 vectors and determine whether 2 vectors are perpendicular to each other, in that sense, the vector theorem has no flaws at all. There is still a struggle of the value obtained for |A||B|cosC, they said it is a scalar value, scalar value in the sense of what? two vectors of the same magnitude and direction cannot even add up with each other by using the dot product.

Answer :

So, that means geometrically, my two vectors are going more or less in the same direction. They make an acute angle. It's going to be zero if the angle is exactly 90°, OK, because that's when the cosine will be zero. And, it will be negative if the angle is more than 90°. So, that means they go, however, in opposite directions. So, that's basically one way to think about what dot product measures. It measures how much the two vectors are going along each other.

Quoted from MIT OCW 18.02 fall 2007

The meaning of DOT PRODUCT of 2 2D vectors, to determine how much a component A goes along component B, normally we compute this by setting component B as a unit vector so it does not causes any disturbance in the process.

Application of DOT PRODUCT
1) obtain angle between 2 vectors
2) Check whether 2 vectors are perpendicular to each other
3) determine the how much does component of Vector A goes along Vector B
4) Area of a parallelogram = |A| |B| sinC proven by base x height, | A || B | sinC = | A' || B | cos (90-C) = A ' . B = det (A, B)

Det(A,B,C) = + - Volume of parallelepipe

CROSS PRODUCT - L02 33.00
cross product of 2 vectors in 3D space
Definition : A x B = | i j k |
| a b c |
| d e f |
Definition : | A x B | which is the length of vector A and B = area of parallelogram formed by vector A and B

From 8.01 L03 fall 1999, Professor Lewin mention that the magnitude of The cross product is simply | A || B | sin C and the direction is just perpendicular to vector A and B, that clears the mystery of why length of A x B will equal to area of a parallelogram.


Application of CROSS PRODUCT
1) |A x B| = Area of parallelogram
2) dir (A x B) = Combined VectorAB that weirdly follows the right hand rule and goes along the third axis that is perpendicular to vector A and vector B.
Right Hand Rule => a) hand point along direction of A b) fingers point towards the direction of B c) the direction the thumb points is dir (A x B).

Geometrically, drawing A B C and using base x height, we get A . (B x C)
finally we check that,

Det(A,B,C) = A . (B x C)
for the full explaination refer to
18.02 L02 52.28

Extending the Applications
if the third vector C is parallel to the plane of AB we get,
C x (A . B) = 0
means that the volume obtained by taking the triple scalar product is zero.
N = A x B , which N is the vector perpendicular to the plane AB
C . ( N ) = 0, which C has no components, or is perpendicular along N

Matrix that turns the plane 90degrees clockwise or anticlockwise
| 0 -1 |
| 1 0 |