Answer
The dot product can be used to test if the two vectors are orthogonal or not; that is, perpendicular to each other.
$a \cdot b =0$ iff a and b are orthogonal.
If the magnitude of $a$ (that is, $|a|$) and the magnitude of $b$ (that is, $|b|$) is known and the angle between $a$ and $b$ (that is, $\theta$) is to be known, the product of the magnitudes and the cosine of the angle is the dot product of $a$ and $b$.
$a \cdot b= |a| |b| cos \theta$
To project one vector onto another -- in short, to find the component of the first vector parallel to the second -- the dot product is very useful.
The projection of $u$ onto $v$ is given as:
$\dfrac{u \cdot v}{|v|^2}v$
Work Step by Step
The dot product can be used to test if the two vectors are orthogonal or not; that is, perpendicular to each other.
$a \cdot b =0$ iff a and b are orthogonal.
If the magnitude of $a$ (that is, $|a|$) and the magnitude of $b$ (that is, $|b|$) is known and the angle between $a$ and $b$ (that is, $\theta$) is to be known, the product of the magnitudes and the cosine of the angle is the dot product of $a$ and $b$.
$a \cdot b= |a| |b| cos \theta$
To project one vector onto another -- in short, to find the component of the first vector parallel to the second -- the dot product is very useful.
The projection of $u$ onto $v$ is given as:
$\dfrac{u \cdot v}{|v|^2}v$
