Question

Projection

Evaluate $v·{proj}_v\mathbf{u}$

Short answer

The result on evaluating is $v.proju=u·v$

Step-by-step solution

Step 1. Given information

A force is given as $\mathbf{F}=4\mathbf{i}-7\mathbf{j}$ and displacement of body is 4

Step 2. Concept used

Projection of u onto v is given by:

${proj}_{v}u=\left(\frac{u·v}{|v{|}^2}\right)v$

If $u=\left(a_1,b_1\right)\&v=\left(a_2,b_2\right)$ then,

$u.v=a_1{a}_2+b_1{b}_2$and $v=\sqrt{a_2^2+b_2^2}$

Use the property of projection of vector u on v

Step 3. Calculation

The projection of u on v is as follows

${proj}_{v}u=\frac{u.v}{|v{|}^2}v$

We can write as follows,

$\begin{array}{l}v\cdot {\mathrm{proj}}_{v}u=v\cdot \left(\frac{u\cdot v}{|v{|}^2}v\right)\\ v\cdot {\mathrm{proj}}_{v}u=\frac{u\cdot v}{|v{|}^2}v\cdot v\\ v\cdot {\mathrm{proj}}_{v}u=\frac{u\cdot v}{|v{|}^2}|v{|}^2\\ v\cdot {\mathrm{proj}}_{z}u=u\cdot v\end{array}$