Scalar product of vectors

The angle between vectors

Let two non-collinear vectors be given $\vec{a}$ and $\vec{b}$. Let's set aside both of these vectors from the same point in the plane. The angle between two vectors is the minimum angle from 0° до 180°, by which one of them must be rotated so that the directions of the vectors coincide. If the vectors are co-directional, then the angle between them is 0°, if they are oppositely directed, then the angle between them is 180°. If one of the vectors or both vectors are zero, then the angle between them is considered to be zero. The angle between vectors $\vec{a}$ and $\vec{b}$ is denoted by $\widehat{\vec{a},\vec{b}}$.

Scalar product of vectors and its properties

The scalar product of two vectors is a number equal to the product of the lengths of these vectors by the cosine of the angle between them: $$\vec{a}\cdot \vec{b}=|\vec{a}|\cdot |\vec{b}|\cdot \mathrm{cos(}\widehat{\vec{a},\vec{b}})$$ If the vectors $\vec{a}$ and $\vec{b}$ are perpendicular, then $\mathrm{cos(}\widehat{\vec{a},\vec{b}}\mathrm{)\; =\; 0}$ and hence, $\vec{a}\cdot \vec{b}=\; 0$. Conversely, if $\vec{a}$ and $\vec{b}$ are nonzero, and $\vec{a}\cdot \vec{b}=\; 0$, then it follows from the definition of the scalar product that $\mathrm{cos(}\widehat{\vec{a},\vec{b}}\mathrm{)\; =\; 0}$.
Thus, a necessary and sufficient condition for the perpendicularity of two vectors is that their scalar product is equal to zero.
From the definition of the scalar product for non-zero vectors it follows that if the angle between the vectors is acute, then the scalar product is a positive number; if the angle is obtuse, then it is negative.
If the angle between the vectors is zero, then $\vec{a}\cdot \vec{b}=|\vec{a}|\cdot |\vec{b}|$. It follows that $\vec{a}\cdot \vec{a}=|\vec{a}|$².
If the angle between vectors 180°, then $\vec{a}\cdot \vec{b}=\; -|\vec{a}|\cdot |\vec{b}|$.
Obviously, for any vectors $\vec{a}$, $\vec{b}$, $\vec{c}$ and an arbitrary number q, the scalar product has the following properties:
1)  $\vec{a}\cdot \vec{b}=\vec{b}\cdot \vec{a}$;
2)  $(\vec{a}+\vec{b})\cdot \vec{c}=\vec{a}\cdot \vec{c}+\vec{b}\cdot \vec{c}$;
3)  $(q\vec{a})\cdot \vec{b}=q(\vec{a}\cdot \vec{b})$.