We defined the dot product of two vectors, [math]\vec{u}[/math] and [math]\vec{v}[/math], which form an angle of [math]\theta[/math] between them as:[br][math]\vec{u}\cdot\vec{v}=\left|\vec{u}\right|\left|\vec{v}\right|\cos\theta[/math]

What if we didn't know the angle between the vectors, but we had their component form? Could we still find their dot product?[br][br]Suppose we are given two vectors [math]\vec{u}[/math] and [math]\vec{v}[/math]. How could we express their component forms? Or, equivalently, how could we find the coordinates of their heads, given that their tails are at the origin?[br][br]Recall that a point on the unit circle has coordinates [math]\left(\cos\theta,\sin\theta\right)[/math] where [math]\theta[/math] is in standard position. This corresponds to a unit vector [math]\left\langle\cos\theta,\sin\theta\right\rangle[/math], but they it needs to be scaled to the appropriate magnitude. So using the angles [math]\alpha[/math] and [math]\beta[/math], we can get that:[br][br][math]\vec{u}=\left\langle x_1,y_1\right\rangle=\left\langle\left|\vec{u}\right|\cos\alpha,\left|\vec{u}\right|\sin\alpha\right\rangle[/math][br][math]\vec{v}=\left\langle x_2,y_2\right\rangle=\left\langle\left|\vec{v}\right|\cos\beta,\left|\vec{v}\right|\sin\beta\right\rangle[/math][br][br]Now consider the c[color=#333333]osine difference formula:[br][/color][math]\cos\left(\alpha-\beta\right)=\cos\alpha\cos\beta+\sin\alpha\sin\beta[/math][br][br]Multiply both sides by [math]\left|\vec{u}\right|\left|\vec{v}\right|[/math][br][math]\left|\vec{u}\right|\left|\vec{v}\right|\cos\left(\alpha-\beta\right)=\left|\vec{u}\right|\left|\vec{v}\right|\left(\cos\alpha\cos\beta+\sin\alpha\sin\beta\right)[/math][br][br][math]\left|\vec{u}\right|\left|\vec{v}\right|\cos\left(\alpha-\beta\right)=\left|\vec{u}\right|\cos\alpha\left|\vec{v}\right|\cos\beta+\left|\vec{u}\right|\sin\alpha\left|\vec{v}\right|\sin\beta[/math][br][br]Notice that the left side matches our definition of dot product, as [math]\alpha-\beta[/math] is the angle between the two vectors [math]\vec{u}[/math] and [math]\vec{v}[/math]. On the right side, we actually have each of the horizontal and vertical components of vectors [math]\vec{u}[/math] and [math]\vec{v}[/math]. Rewriting we have:[br]

[math]\vec{u}\cdot\vec{v}=x_1x_2+y_1y_2[/math][br][br]This is the dot product of two vectors given in component form: [math]\vec{u}=\left\langle x_1,y_1\right\rangle[/math] and [math]\vec{v}=\left\langle x_2,y_2\right\rangle[/math][br]We take the sum of the products of their corresponding components.