[b]Dot Products[/b][br][br]Given any two vectors [math]\vec{u}[/math] and [math]\vec{v}[/math] in [math]\mathbb{R}^2[/math] or [math]\mathbb{R}^3[/math], we define their [b]dot product[/b] as follows:[br][br][math]\vec{u}\cdot \vec{v}=|\vec{u}| |\vec{v}| \cos\theta [/math][br][br]when both vectors are non-zero and [math]\theta[/math] is the angle in the range from [math]0[/math] to [math]\pi[/math] formed by translating two non-zero vectors such that their tails meet at a point. If either [math]\vec{u}=0[/math] or [math]\vec{v}=0[/math], [math]\theta[/math] is not defined and [math]\vec{u}\cdot \vec{v}[/math] is defined to be [math]0[/math].[br][br][u]Remark[/u]: From the definition, we have [math]\vec{u}\cdot\vec{v}=\vec{v}\cdot\vec{u}[/math].[br][br]In the applet below, bring the two vectors together in the plane to find out their dot product.[br]

[u]Definition[/u]: Two vectors are [b]orthogonal[/b] if their dot product is zero i.e. if two vectors are orthogonal, they are either both non-zero and perpendicular to each other, or at least one of the vectors is a zero vector. [br][br][u]Remark[/u]: Zero vector is orthogonal to any vector.