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What is the physical interpretation of the dot/inner/scalar product of two vectors?

See, if we multiply two scalars like 2*3 we say two times three is six. I also do understand multiplication of vectors with scalars, ie. say $x$ times a vector $\vec{a}$. But not for two vectors being multiplied together to form a scalar.

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There are two kinds of vector multiplications-

Dot (Scalar) Product:

The dot product of two vectors gives a scalar, that means only the magnitude is left, no direction. Mathematically, it is equal to the product of the magnitude of two vectors times the cosine of the angle between the two. ie. $$\vec{v} \cdot \vec{u}=|\vec{v}||\vec{u}|Cos\theta$$

The geometric interpretation: The dot product of $\vec{a}$ with unit vector $\hat{u}$, denoted $\vec{a}⋅\hat{u}$, is defined to be the projection of $\vec{a}$ in the direction of $\vec{a}$, or the amount that $\vec{a}$ is pointing in the same direction as unit vector $\hat{u}$. Let's assume for a moment that $\vec{a}$ and $\hat{u}$ are pointing in similar directions. Then, you can imagine $\vec{a}⋅\hat{u}$ as the length of the shadow of $\vec{a}$ onto $\hat{u}$ if their tails were together and the sun was shining from a direction perpendicular to $\hat{u}$. By forming a right triangle with $\vec{a}$ and this shadow, you can use geometry to calculate that $$\vec{a}⋅\hat{u}=|\vec{a}|Cos\theta$$

enter image description here

Cross (Vector) Product:

The cross product of two vectors gives a vector, that means the answer has a magnitude and a direction. The magnitude of the resultant vector is given by the product of the magnitude of the two vectors times the sine of the angle between them. The cross product is always perpendicular to both vectors, and has magnitude zero when the vectors are parallel and maximum magnitude when they are perpendicular. $$\vec{v} \times \vec{u}=|\vec{v}||\vec{u}|Sin\theta \hat{r}$$ where $\hat{r}$ is the unit vector in the direction of the resultant vector. The direction of this can be found out using the gif below or the right hand thumb rule.

enter image description here Right hand thumb rule

The geometrical interpretation: The magnitude of the cross product can be interpreted as the positive area of the parallelogram having $\vec{v}$ and $\vec{u}$ as sides.

enter image description here

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