Question

Projecting one vector onto another: Show that the formula for the projection of a vector v onto a nonzero vector u is given by$pro{j}_{\mathit{u}}\mathit{v}\mathbf{}\mathbf{=}\mathbf{}\frac{\mathbf{u}\mathbf{.}\mathbf{v}}{\left|\left|u\right|\right|}\mathbf{,}\mathbf{}whereu\ne 0.$

Short answer

$$\begin{gathered} Dotproduct: \\ \mathit{u}\mathbf{.}\mathit{v}=\left|\mathit{u}\right|\left|\mathit{v}\right|\cos \theta , \\ pro{j}_{\mathbf{u}}\mathit{v}=\left|\mathit{v}\right|\cos \theta , \\ Solvingabovetwoequationsweget, \\ pro{j}_{\mathbf{u}}\mathit{v}=\frac{\mathit{u}\mathbf{.}\mathit{v}}{\left|\left|\mathit{u}\right|\right|}. \end{gathered}$$

Step-by-step solution

Step 1. Given Information.

Given two vectors uand vwhere u is a nonzero vector.

Step 2. Dot product.

$\mathit{u}\mathbf{.}\mathit{v}=\left|u\right|.\left|v\right|\cos \theta$

Step 3. Proof

$$\begin{gathered} Letsupposewehaveavector\mathit{v}andanothervector\mathit{u}andthesevectors \\ makeanangleof\theta betweenthem, \\ Now,thewetakethecomponentofthe\mathit{v}alongthe\mathit{u}weget \\ com{p}_{\mathbf{u}}\mathit{v}=\left|\mathit{v}\right|\cos \theta , \\ Nowtheprojectionisthesameasthecomponentwecanwrite, \\ pro{j}_{\mathbf{u}}\mathit{v}=com{p}_{\mathbf{u}}\mathit{v}=\left|\mathit{v}\right|\cos \theta \\ Nowmultiplydividebythemagnitudeof\mathit{u}vectorinthenumeratorand \\ denominatorintheRHS,weget, \\ pro{j}_{\mathbf{u}}\mathit{v}=\frac{\left|\mathit{v}\right|\left|\mathit{u}\right|\cos \theta }{\left|\mathit{u}\right|}, \\ Nowfromdotproductweget, \\ pro{j}_{\mathbf{u}}\mathit{v}=\frac{\mathit{u}\mathbf{.}\mathit{v}}{\left|\left|\mathit{u}\right|\right|}. \end{gathered}$$