Orthogonal Projections — Practice Problems and Worked Solutions

The formulas you need here

18 worked problems

Every step is generated and checked by a computer algebra system.

1. 1.

   Project v onto the line spanned by u.

   $$v=\left[\begin{array}{c}2\\ 3\end{array}\right],u=\left[\begin{array}{c}1\\ 0\end{array}\right]$$

   Answer: $\left[\begin{array}{c}2\\ 0\end{array}\right]$

   Step-by-step solution

   1. $${\mathrm{proj}}_u(v)=\frac{v\cdot u}{u\cdot u}u$$
   2. $$=\left[\begin{array}{c}2\\ 0\end{array}\right]$$
2. 2.

   Project v onto the line spanned by u.

   $$v=\left[\begin{array}{c}3\\ 4\end{array}\right],u=\left[\begin{array}{c}0\\ 2\end{array}\right]$$

   Answer: $\left[\begin{array}{c}0\\ 4\end{array}\right]$

   Step-by-step solution

   1. $${\mathrm{proj}}_u(v)=\frac{v\cdot u}{u\cdot u}u$$
   2. $$=\left[\begin{array}{c}0\\ 4\end{array}\right]$$
3. 3.

   Project v onto the line spanned by u.

   $$v=\left[\begin{array}{c}2\\ 1\end{array}\right],u=\left[\begin{array}{c}1\\ 1\end{array}\right]$$

   Answer: $\left[\begin{array}{c}\frac{3}{2}\\ \frac{3}{2}\end{array}\right]$

   Step-by-step solution

   1. $${\mathrm{proj}}_u(v)=\frac{v\cdot u}{u\cdot u}u$$
   2. $$=\left[\begin{array}{c}\frac{3}{2}\\ \frac{3}{2}\end{array}\right]$$
4. 4.

   Project v onto the line spanned by u.

   $$v=\left[\begin{array}{c}4\\ -2\end{array}\right],u=\left[\begin{array}{c}1\\ -1\end{array}\right]$$

   Answer: $\left[\begin{array}{c}3\\ -3\end{array}\right]$

   Step-by-step solution

   1. $${\mathrm{proj}}_u(v)=\frac{v\cdot u}{u\cdot u}u$$
   2. $$=\left[\begin{array}{c}3\\ -3\end{array}\right]$$
5. 5.

   Project v onto the line spanned by u.

   $$v=\left[\begin{array}{c}1\\ 2\\ 3\end{array}\right],u=\left[\begin{array}{c}1\\ 0\\ 1\end{array}\right]$$

   Answer: $\left[\begin{array}{c}2\\ 0\\ 2\end{array}\right]$

   Step-by-step solution

   1. $${\mathrm{proj}}_u(v)=\frac{v\cdot u}{u\cdot u}u$$
   2. $$=\left[\begin{array}{c}2\\ 0\\ 2\end{array}\right]$$
6. 6.

   Project v onto the line spanned by u.

   $$v=\left[\begin{array}{c}3\\ -1\\ 2\end{array}\right],u=\left[\begin{array}{c}1\\ 2\\ 0\end{array}\right]$$

   Answer: $\left[\begin{array}{c}\frac{1}{5}\\ \frac{2}{5}\\ 0\end{array}\right]$

   Step-by-step solution

   1. $${\mathrm{proj}}_u(v)=\frac{v\cdot u}{u\cdot u}u$$
   2. $$=\left[\begin{array}{c}\frac{1}{5}\\ \frac{2}{5}\\ 0\end{array}\right]$$
7. 7.

   Project v onto the line spanned by u.

   $$v=\left[\begin{array}{c}4\\ 2\end{array}\right],u=\left[\begin{array}{c}1\\ 0\end{array}\right]$$

   Answer: $\left[\begin{array}{c}4\\ 0\end{array}\right]$

   Step-by-step solution

   1. $${\mathrm{proj}}_u(v)=\frac{v\cdot u}{u\cdot u}u$$
   2. $$=\left[\begin{array}{c}4\\ 0\end{array}\right]$$
8. 8.

   Project v onto the line spanned by u.

   $$v=\left[\begin{array}{c}5\\ 5\end{array}\right],u=\left[\begin{array}{c}0\\ 1\end{array}\right]$$

   Answer: $\left[\begin{array}{c}0\\ 5\end{array}\right]$

   Step-by-step solution

   1. $${\mathrm{proj}}_u(v)=\frac{v\cdot u}{u\cdot u}u$$
   2. $$=\left[\begin{array}{c}0\\ 5\end{array}\right]$$
9. 9.

   Project v onto the line spanned by u.

   $$v=\left[\begin{array}{c}3\\ 1\end{array}\right],u=\left[\begin{array}{c}2\\ 1\end{array}\right]$$

   Answer: $\left[\begin{array}{c}\frac{14}{5}\\ \frac{7}{5}\end{array}\right]$

   Step-by-step solution

   1. $${\mathrm{proj}}_u(v)=\frac{v\cdot u}{u\cdot u}u$$
   2. $$=\left[\begin{array}{c}\frac{14}{5}\\ \frac{7}{5}\end{array}\right]$$
10. 10.

    Project v onto the line spanned by u.

    $$v=\left[\begin{array}{c}6\\ -3\end{array}\right],u=\left[\begin{array}{c}1\\ -1\end{array}\right]$$

    Answer: $\left[\begin{array}{c}\frac{9}{2}\\ -\frac{9}{2}\end{array}\right]$

    Step-by-step solution

    1. $${\mathrm{proj}}_u(v)=\frac{v\cdot u}{u\cdot u}u$$
    2. $$=\left[\begin{array}{c}\frac{9}{2}\\ -\frac{9}{2}\end{array}\right]$$
11. 11.

    Project v onto the line spanned by u.

    $$v=\left[\begin{array}{c}2\\ 4\\ 1\end{array}\right],u=\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right]$$

    Answer: $\left[\begin{array}{c}2\\ 0\\ 0\end{array}\right]$

    Step-by-step solution

    1. $${\mathrm{proj}}_u(v)=\frac{v\cdot u}{u\cdot u}u$$
    2. $$=\left[\begin{array}{c}2\\ 0\\ 0\end{array}\right]$$
12. 12.

    Project v onto the line spanned by u.

    $$v=\left[\begin{array}{c}1\\ 3\\ -2\end{array}\right],u=\left[\begin{array}{c}0\\ 1\\ 1\end{array}\right]$$

    Answer: $\left[\begin{array}{c}0\\ \frac{1}{2}\\ \frac{1}{2}\end{array}\right]$

    Step-by-step solution

    1. $${\mathrm{proj}}_u(v)=\frac{v\cdot u}{u\cdot u}u$$
    2. $$=\left[\begin{array}{c}0\\ \frac{1}{2}\\ \frac{1}{2}\end{array}\right]$$
13. 13.

    Project v onto the line spanned by u.

    $$v=\left[\begin{array}{c}5\\ 2\end{array}\right],u=\left[\begin{array}{c}1\\ 2\end{array}\right]$$

    Answer: $\left[\begin{array}{c}\frac{9}{5}\\ \frac{18}{5}\end{array}\right]$

    Step-by-step solution

    1. $${\mathrm{proj}}_u(v)=\frac{v\cdot u}{u\cdot u}u$$
    2. $$=\left[\begin{array}{c}\frac{9}{5}\\ \frac{18}{5}\end{array}\right]$$
14. 14.

    Project v onto the line spanned by u.

    $$v=\left[\begin{array}{c}2\\ 6\end{array}\right],u=\left[\begin{array}{c}3\\ 1\end{array}\right]$$

    Answer: $\left[\begin{array}{c}\frac{18}{5}\\ \frac{6}{5}\end{array}\right]$

    Step-by-step solution

    1. $${\mathrm{proj}}_u(v)=\frac{v\cdot u}{u\cdot u}u$$
    2. $$=\left[\begin{array}{c}\frac{18}{5}\\ \frac{6}{5}\end{array}\right]$$
15. 15.

    Project v onto the line spanned by u.

    $$v=\left[\begin{array}{c}4\\ 0\\ 3\end{array}\right],u=\left[\begin{array}{c}1\\ 1\\ 0\end{array}\right]$$

    Answer: $\left[\begin{array}{c}2\\ 2\\ 0\end{array}\right]$

    Step-by-step solution

    1. $${\mathrm{proj}}_u(v)=\frac{v\cdot u}{u\cdot u}u$$
    2. $$=\left[\begin{array}{c}2\\ 2\\ 0\end{array}\right]$$
16. 16.

    Project v onto the line spanned by u.

    $$v=\left[\begin{array}{c}1\\ 1\\ 4\end{array}\right],u=\left[\begin{array}{c}2\\ 0\\ 1\end{array}\right]$$

    Answer: $\left[\begin{array}{c}\frac{12}{5}\\ 0\\ \frac{6}{5}\end{array}\right]$

    Step-by-step solution

    1. $${\mathrm{proj}}_u(v)=\frac{v\cdot u}{u\cdot u}u$$
    2. $$=\left[\begin{array}{c}\frac{12}{5}\\ 0\\ \frac{6}{5}\end{array}\right]$$
17. 17.

    Project v onto the line spanned by u.

    $$v=\left[\begin{array}{c}7\\ 1\end{array}\right],u=\left[\begin{array}{c}1\\ 1\end{array}\right]$$

    Answer: $\left[\begin{array}{c}4\\ 4\end{array}\right]$

    Step-by-step solution

    1. $${\mathrm{proj}}_u(v)=\frac{v\cdot u}{u\cdot u}u$$
    2. $$=\left[\begin{array}{c}4\\ 4\end{array}\right]$$
18. 18.

    Project v onto the line spanned by u.

    $$v=\left[\begin{array}{c}3\\ 3\\ 3\end{array}\right],u=\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right]$$

    Answer: $\left[\begin{array}{c}3\\ 3\\ 3\end{array}\right]$

    Step-by-step solution

    1. $${\mathrm{proj}}_u(v)=\frac{v\cdot u}{u\cdot u}u$$
    2. $$=\left[\begin{array}{c}3\\ 3\\ 3\end{array}\right]$$

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• Matrix Inverses
• Linear Independence
• Rank & Basis
• Linear Transformations
• Eigenvalues