# How do I find the dot product of vectors v =5i-2j and w=3i+4j?

Dec 31, 2014

$\vec{v} \cdot \vec{w} = 7$
The dot product is a scalar obtained by multiplying the corresponding components of the two vectors and adding (algebraically) the results.
So you have:
$\vec{v} \cdot \vec{w} = \left(5 \cdot 3\right) + \left(- 2 \cdot 4\right) = 15 - 8 = 7$

You may check your result by plotting your vectors and using the alternative definition of dot product:

$\vec{v} \cdot \vec{w} = | \vec{v} | \cdot | \vec{w} | \cdot \cos \left(\theta\right)$
i.e.: the product of the modulus of the vectors times de cosine of the angle between them.

After some inverse trigonometry and Pitagora's Theorem I got:
$| \vec{v} | = 5.4$
$| \vec{w} | = 5$
|theta_v|=21.8°
|theta_w|=53.1°
and theta=21.8°+53.1°=74.9°
so that:
vecv*vecw=5*5.4*cos(74.9°)=7