Question

Vector Projection of $v$ onto $v$

(a) Calculate ${proj}_{\mathbf{v}}\mathbf{u}$

(b) Resolve $u$ into $u1$ and $u2$, where $u1$ is parallel to $v$ and $v2$ is orthogonal to $v$

$\mathbf{u}=⟨7,-4⟩,\mathbf{v}=⟨2,1⟩$

Short answer

1. The projection is ${proj}_{v}u=⟨4,2⟩$
2. By resolving u into u1 and u2 we get,$u_1=⟨4,2⟩,u_2=⟨3,-6⟩$

Step-by-step solution

a.Step 1. Given information

Vectors are given as-

$\mathbf{u}=⟨7,-4⟩,\mathbf{v}=⟨2,1⟩$

Step 2. Concept used

If the vector u is resolved into u1 and u2 , where u1 is parallel to v and u2 is orthogonal to v , then

$\begin{array}{l}{u}_1={\mathrm{Proj}}_{v}u\\ u_2=u-{\mathrm{Proj}}_{v}u\end{array}$

Projection of u onto v is given by:

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

Vector scaler product or Dot product:

$a·b=|a\Vert b|\cos \theta$

Use the formula of projection of vector.

Step 3. Calculation

Projection of u on $v$,

$\begin{array}{l}{\mathrm{proj}}_{v}u=\left\{\frac{u\cdot v}{|v{|}^2}\right\}v\\ =\left(\frac{⟨7,-4⟩\cdot ⟨2,1⟩}{\left(2^2+1^2\right)}\right)⟨2,1⟩\\ =\left(\frac{14-4}{(4+1)}\right)⟨2,1⟩\\ =\left(\frac{10}{5}\right)⟨2,1⟩\\ =2⟨2,1⟩\\ =⟨4,2⟩\end{array}$

b.Step 1. Given information

Vectors are given as-

$\mathbf{u}=⟨7,-4⟩,\mathbf{v}=⟨2,1⟩$ &

${proj}_{v}u=⟨4,2⟩$

Step 2. Concept used

If the vector u is resolved into u1 and u2 , where u1 is parallel to v and u2 is orthogonal to v , then

$\begin{array}{l}{u}_1={\mathrm{Proj}}_{v}u\\ u_2=u-{\mathrm{Proj}}_{v}u\end{array}$

Projection of u onto v is given by:

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

Resolve by using the given definition.

Step 3. Calculation

$\begin{array}{l}Pro{j}_{v}u=⟨4,2⟩\\ u_1=pro{j}_{v}u\\ u_1=⟨4,2⟩\\ u_2=u-{\mathrm{Proj}}_{v}u\\ u_2=⟨7,-4⟩-⟨4,2⟩\\ u_2=⟨7-4,-4-2⟩\\ u_2=⟨3,-6⟩\end{array}$