why do we use cosine as the expression of vector dot product?
When we do vector products, we use two different methods. One is the vector dot product, another is vector cross product. The equation of the vector dot product is $$\textbf A \cdot \textbf B =|\textbf A| | \textbf B| \cos\theta,$$ where $\theta$ is the angle between the vectors $\textbf A$ and $\textbf B$.
Why do we use cosine as the expression?
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$\begingroup$Let $\vec{\mathbf a} = (x_1, y_1) = (a \cos \alpha, a \sin \alpha)$
Let $\vec{\mathbf b} = (x_2, y_2) = (b \cos \beta, b \sin \beta)$
Then $\theta = |\beta - \alpha|$
By definition,
\begin{align} \vec{\mathbf a} \circ \vec{\mathbf b} &= x_1x_2 + y_1y_2 \\ &= ab(\cos \alpha \cos \beta + \sin \alpha \sin \beta) \\ &= ab \cos(\beta - \alpha)\\ &= ab \cos \theta \end{align}
(Note $\cos(\theta) = \cos(-\theta)$)
$\endgroup$ 1 $\begingroup$The dot product of two vectors $A$ and $B$ is just the product of the magnitude of one vector with the scalar projection of the other one on itself. Hence the $cos$ term. Also, note that the $cos$ function is greater for smaller angles, and lesser for larger ones, just like the length of the projection. (both are the same thing, actually.)
$\endgroup$ 1 $\begingroup$The dot product is defined in that way. Note that $cos\theta$ is a suitable function; since by the Schwarz inequality: $$|\mathbf{A} \cdot \mathbf{B}| \leq |\mathbf{A}| |\mathbf{B}|$$ and thus the dot product ranges continuosly between -1 and 1, as $cos \theta$ for $\theta \in [0,\pi] $.
$\endgroup$ $\begingroup$This is entirely determined by what we consider a rotation in the plane.
Let $u, v$ be two unit vectors. Let the angle between them be $\theta$, and we can naturally write $v$ as
$$v = u \cos \theta + u_\perp \sin \theta$$
where $u_\perp$ is a unit vector perpendicular to $u$. Then clearly, the dot product is $u \cdot v = \cos \theta$.
But, if you're not in a Euclidean plane anymore, this relationship no longer holds. For example, in a Lorentzian space, instead of cosine and sine, we get hyperbolic functions instead:
$$v = u \cosh \theta + u_\perp \sinh \theta$$
And the dot product is $u \cdot v = \cosh \theta$. A physicist should recognize that this $\theta$ is the "rapidity", and that the form of $v$ given here is exactly that of any Lorentz boost.
So the reason we use sine and cosine in Euclidean space is because they are dictated by the use of sine and cosine in rotations. In other spaces, with different rotation operators, you use the functions that are associated with those rotations instead.
$\endgroup$ 2 $\begingroup$In dot product we use cos theta because in this type of product 1.) One vector is the projection over the other. 2.) The distance is covered along one axis or in the direction of force and there is no need of perpendicular axis or sin theta. In cross product the angle between must be greater than 0 and less than 180 degree it is max at 90 degree. let take the example of torque if the angle between applied force and moment arm is 90 degree than torque will be max. That's why we use cos theta for dot product and sin theta for cross product.
$\endgroup$ $\begingroup$Because here cosine is responsible for doing any type of work e.g work done cosine is use . Here one axis is use for work done I.e x -axis and for x -axis we are using cosine
$\endgroup$ $\begingroup$Cosine is used to make both the vectors point in same direction. For dot product we require both the vectors to point in same direction and cosine does so by projecting one vector in the same direction as other.
$\endgroup$ $\begingroup$It is actually the definition of the dot product of two vectors.
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