# What is the difference between the cross-product and the dot-product of two vectors? or If a force of 10 Newtons at an angle of 30^circ above horizontal is used to drag a block 6 meters, how much work is done?

Apr 29, 2015

The Cross Product is a vector
If A and B are vectors
then $A \times B$ is a vector perpendicular to both A and B (that is perpendicular to the plane containing A and B if A and B are two dimensional vectors in 3D space) with magnitude equal to the area of the parallelogram with sides A and B

$A \times B = | \setminus | A | \setminus | \cdot | \setminus | B | \setminus | \cdot \sin \left(\theta\right) \cdot n$
where $| \setminus | V | \setminus |$ is the magnitude of a vector, $V$,
$\theta$ is the angle between $A$ and $B$,
and $n$ is a unit length vector perpendicular to the plane containing $A$ and $B$

The Dot Product is a scalar
If A and B are vectors such that
$A = < {a}_{1} , {a}_{2} , \ldots , {a}_{k} >$
and
$B = < {b}_{1} , {b}_{2} , \ldots , {b}_{k} >$

then
$A \cdot B = {\sum}_{i = 1}^{k} \left({a}_{i} \times {b}_{i}\right)$

it can also be evaluated as
$A \cdot B = | \setminus | A | \setminus | \times | \setminus | B | \setminus | \cdot \cos \left(\theta\right)$
where $\theta$ is the angle between A and B.

Apr 29, 2015

Work is a dot product.
It is given as:
$W = \vec{F} \cdot \vec{s}$
Where:
$\vec{F}$ is a force (vector)
$\vec{s}$ is the displacement (vector)
The result is a SCALAR (basically a quantity without vectorial characteristics) and is measured in Joules (J).
To evaluate it you do:
$W = | \vec{F} | \cdot | \vec{s} | \cdot \cos \left(\theta\right)$
where you multiply the modulus of your vectors times the cosine of the angle between them.
For example:

Hope it helps.