If we have two vectors [math]\vec{u}[/math] and [math]\vec{v}[/math] that are in the same direction, then their [b]dot product[/b] is simply the product of their magnitudes: [math]\vec{u}\cdot\vec{v}=\left|\vec{u}\right|\left|\vec{v}\right|[/math]. To see this above, drag the head of [math]\vec{v}[/math] to make it parallel to [math]\vec{u}[/math].[br][br]If the two vectors are not in the same direction, then we can find the component of vector [math]\vec{v}[/math] that is parallel to vector [math]\vec{u}[/math], which we can call [math]\vec{w}[/math]. and take the product of the magnitudes of [math]\vec{u}[/math] and [math]\vec{w}[/math]: [math]\vec{u}\cdot\vec{v}=\left|\vec{u}\right|\left|\vec{w}\right|[/math][br][br]But how can we find [math]\left|\vec{w}\right|[/math]? If the angle between vectors [math]\vec{u}[/math] and [math]\vec{v}[/math] is [math]\theta[/math], then we can see that [math]\cos\theta=\frac{\left|\vec{w}\right|}{\left|\vec{v}\right|}[/math], so [math]\left|\vec{w}\right|=\left|\vec{v}\right|\cos\theta[/math].[br][br][size=100]Therefore, in general, we have that the [b]dot product[/b] o[/size]f [math]\vec{u}[/math] and [math]\vec{v}[/math] is:[br][math]\vec{u}\cdot\vec{v}=\left|\vec{u}\right|\left|\vec{v}\right|\cos\theta[/math][br]where [math]\theta[/math] is the angle between the two vectors [math]\vec{u}[/math] and [math]\vec{v}[/math].