Scalar product and vector product

[prashant singh]

why do we take cross product of A X B as a line normal to the plane which contains A and B. I also need a proof of A.B = |A||B|cos(theta), I have seen many proves but they have used inter product ,A.A = |A|^2, which is a result of dot product with angle = 0, we can't use this too prove the dot product formula.
First one is more important please help.

 

[blue_leaf77]

why do we take cross product of A X B as a line normal to the plane which contains A and B.

You asked why, well it's because a vector product is defined that way. It's defined such that the result of the product is perpendicular to any linear combination of A and B.

I also need a proof of A.B = |A||B|cos(theta)

Consider the cosine rules for the addition and the difference between two vectors.
$$
|\vec{A}+\vec{B}|^2 = |\vec{A}|^2 + |\vec{B}|^2 + 2|\vec{A}||\vec{B}| \cos\theta \\
|\vec{A}-\vec{B}|^2 = |\vec{A}|^2 + |\vec{B}|^2 - 2|\vec{A}||\vec{B}| \cos\theta \\
$$

 

[FactChecker]

Regarding the first question: We need some fundamental operation that reflects rotations in R³ from direction A to direction B. A nice operation would be linear in both A and B and would be anti-commutative ( AxB = -BxA ). The definition of the cross product fits the bill.

 

[pasmith]

In euclidean 2D space, if $\mathbf{a} = (a \cos \phi, a \sin \phi)$ and $\mathbf{b} = (b \cos \alpha, b \sin \alpha)$ then by basic trigonometry $$\mathbf{a} \cdot \mathbf{b} = (a \cos \phi, a \sin \phi) \cdot (b \cos \alpha, b \sin \alpha) = ab \cos \phi \cos\alpha + ab \sin \phi \sin \alpha = ab \cos(\phi - \alpha),$$ or $$\mathbf{a} \cdot \mathbf{b} = \|\mathbf{a}\|\|\mathbf{b}\| \cos\theta$$ where $\theta = \phi - \alpha$ is the angle between $\mathbf{a}$ and $\mathbf{b}$.

In arbitrary inner product spaces, $\|a\| = (a \cdot a)^{1/2}$ is the definition of $\|a\|$, and after one has proven from basic properties of the inner product that $|a \cdot b| \leq \|a\|\|b\|$ one can then define $\theta$ by $a \cdot b = \|a\|\|b\| \cos \theta$.

 

[prashant singh]

So you are saying that ||a|| = (a.a)^1/2 is the defination given by the founders and there is no proof for this . But why there is no proof .

 

[SammyS]

So you are saying that ||a|| = (a.a)^1/2 is the defination given by the founders and there is no proof for this . But why there is no proof .

In general, there is no proof for a definition. That would not be logical.

The definition tells the some meaning of something.