Question about the dot product

Question

I understand the geometric meaning of cross product but not getting with what dot product of two vectors actually implies geometrically.

Answer 1

Geometrically, the dot product is the product of the magnitudes of the vectors multiplied by the cosine of the angle between them. Since we are only speaking about magnitudes and magnitude is a scalar (has no direction), the dot product is also called the scalar product. The dot product of two vectors is a scalar, it has only magnitude and no direction.

Answer 2

The dot product can be defined as:

$$\vec a\cdot\vec b=|\vec a||\vec b| \cos(\theta)$$

Rewritten as:

$$\cos(\theta)=\frac{\vec a\cdot\vec b}{|\vec a||\vec b|}$$

$$\theta=\arccos\left(\frac{\vec a\cdot\vec b}{|\vec a||\vec b|}\right)$$

we see that the dot product can be used to calculate the angle between two vectors.

Note if $\theta=\frac{\pi}2$, then the dot product is zero, and the two vectors are orthogonal.

Answer 3

Roughly, if $\textbf x$ is a unit vector, then $\textbf x\cdot\textbf y$ measures the projection of $\textbf y$ in the direction of $\textbf x$ (and by symmetry vice-versa). The actual vector being measured then is $(\textbf x\cdot \textbf y)\textbf x$, since it has the right direction ($\textbf x$) and the right magnitude ($\textbf x\cdot\textbf y)$. I think I said that right.

So in some sense, the dot product and the cross product are complementary notions (though there is something of an asymmetry because one yields a scalar and the other a vector).