What are vector products?

Question

What is the difference between dot and cross product?
why 2 of them are needed?
How do we resolve 3 components? ie if it is cos i and sin j what will be for k?

Answer 1

Geometrically:

A dot product can be used to find the angle between two vectors since, by definition, A dot B = |A| |B| cos (theta). In other words, the dot product is equal to the magnitude of the two vectors times the cosine of the angle between them. If you know the vector components, you can rearrange the equation to find the angle between the two vectors. (For a two dimensional vector, you might notice the dot product is basically the cosine difference identity).

A cross product creates a new vector that's perpendicular to the first two vectors. It's magnitude is equal to the area of a parallelogram that would be defined by the first two vectors. i.e - A X B = |A| |B| sin (theta). It's use is completely different than the cross product. You use it to find torques, angular momentum, etc.

If you have only two components of a three dimensinal vector, you can find the third using the Pythagorean Theorem. The Pythagorean Theorem works in any number of dimensions - the two dimensional version you initially learned just had 0 for it's z component, it's fourth dimensional component, it's fifth dimensional component, etc.

For the example you gave, knowing that cos^2(theta) + sin^2(theta) = 1, and that the sum of the squares of the cosines for a three-dimensional vector has to equal 1, you realize there's nothing left for z. It has to equal 0.

A better example would have something for all three components, such as "what angle does the diagonal of a cube have to each of the sides?" The angle would have to be the same to each side, meaning the same angle can be substituted for all three components:

cos^2(theta) + cos^2(theta) + cos^2(theta) = 1^2
3 cos^2(theta) = 1
cos^2(theta) = 1/3
cos(theta) = 1/sqrt(3)
theta = 54.7 degrees

Answer 2

There's like 7 questions here!

1. Dot product is a scalar, cross product is a vector.

2. Because we can. Both have science and engineering applications.

3. If cos i and sin j exist and you are in 3-D space, then k would be 0.

Answer 3

Is this the shady knife company that recruits college kids to try and sell overpriced knife sets to all their family and friends?

If so I'll tell you wha their product is! Poverty for naive job applicants!

Answer 4

Try Wikipedia

Answer 5

cross product of v x w is a vector that is orthogonal on and v and w.