Question

The projection of the vector $\overrightarrow{A}=\hat{i}-2\hat{j}+\hat{k}onthevector\overrightarrow{B}=4\hat{i}-4\hat{j}+7\hat{k}$ is a.

• A. $\frac{19}{9}$
• B. $\frac{38}{9}$
• C. $\frac{8}{9}$
• D. $\frac{4}{9}$

Answer 1

The correct option is C $\frac{8}{9}$

$\overline{A}=i-2j+k$

$B=4i-4j+7k$

${Proj}_{\overline{B}}\left(\overline{A}\right)=\left(\frac{\overline{A}.\overline{B}}{\left|\overline{B}\right|}\right)\frac{\overline{B}}{\left|\overline{B}\right|}$

$\overline{A}=1,-2,1$

$\overline{B}=4,-4,7$

$\overline{A}.\overline{B}=⟨1,-2,1⟩.⟨4,-4,7⟩$

$=(1\times 4)+(-2\times -4)+(1\times 7)$

$=4+8+7$

$=15+4=19$

$\left|\overline{A}\right|=\sqrt{1^2+{(-2)}^2+1}=\sqrt{6}$

$\left|\overline{B}\right|=\sqrt{16+{(-4)}^2+7^2}=\sqrt{16+16+49}$

$=\sqrt{32+49}=\sqrt{81}=9$

${Proj}_{\overline{B}}\overline{A}=\left(\frac{19}{9}\right)\frac{⟨4,-4,7⟩}{9}$

$=\frac{19}{81}⟨4,-4,7⟩$

$=⟨\frac{76}{81},-\frac{76}{81},\frac{133}{81}⟩$

$=\frac{8}{9}$