Question

What exactly do find when we dot and cross two vectors?

I know the properties and the equations for the two but what does the final result actually represent .

Answer 1

As both of them has got different aspects , we have to use both of them.

Explanation:

The dot product or scalar product is an algebraic operation that takes two equal-length sequences of numbers and returns a single number. This operation can be defined either algebraically or geometrically.

The cross product or vector product is a binary operation on two vectors in three-dimensional space and is denoted by the symbol ×. The cross product a × b of two linearly independent vectors a and b is a vector that is perpendicular to both and therefore normal to the plane containing them. It has many applications in mathematics, physics, engineering, and computer programming.

Answer 2

One way of looking at it...

Explanation:

Both dot and cross products actually come from quaternion arithmetic.

Quaternions are the four dimensional, associative, non-commutative division algebra over the reals, extending the Complex numbers.

A complex number is representable as `a+bi` where `a, b` are real numbers and `i` is the imaginary unit.

A quaternion is representable as `a+bi+cj+dk` where `a, b, c, d` are real numbers and `i, j, k` are orthogonal imaginary units satisfying:

• `i^2 = j^2 = k^2 = ijk = -1`

• `ij = k`, `jk = i`, `ki = j`, `ji = -k`, `kj = -i`, `ik = -j`

We can separate a quaternion into scalar and vector parts:

`a+bi+cj+dk = (a, < b, c, d >)`

Then quaternion addition and multiplication can be defined as:

`(a, vec(u)) + (b, vec(v)) = (a+b,color(white)(.) vec(u)+vec(v))`

`(a, vec(u)) (b, vec(v)) = (ab - vec(u) * vec(v),color(white)(.) a vec(v)+b vec(u)+ vec(u) xx vec(v))`

Using this kind of arithmetic, we can both multiply and divide vectors in `4` dimensional space.

Note that if the scalar part of two quaternions is zero then multiplication basically gives you the dot product and cross product simultaneously:

`(0, vec(u)) (0, vec(v)) = (-vec(u) * vec(v),color(white)(.) vec(u) xx vec(v))`

Quaternions are useful in mechanics, though not used so often nowadays.

Search for quaternion on Socratic for related questions.