Dot Products — Practice Problems and Worked Solutions

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17 worked problems

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

1. 1.

   Find the dot product.

   $$u=\left[\begin{array}{c}1\\ 2\end{array}\right],v=\left[\begin{array}{c}3\\ 4\end{array}\right],u\cdot v=?$$

   Answer: $11$

   Step-by-step solution

   1. $$u\cdot v=\sum _i{u}_i{v}_i$$
   2. $$u\cdot v=11$$
2. 2.

   Find the dot product.

   $$u=\left[\begin{array}{c}-1\\ 3\end{array}\right],v=\left[\begin{array}{c}2\\ 5\end{array}\right],u\cdot v=?$$

   Answer: $13$

   Step-by-step solution

   1. $$u\cdot v=\sum _i{u}_i{v}_i$$
   2. $$u\cdot v=13$$
3. 3.

   Find the dot product.

   $$u=\left[\begin{array}{c}2\\ -2\\ 1\end{array}\right],v=\left[\begin{array}{c}4\\ 1\\ 3\end{array}\right],u\cdot v=?$$

   Answer: $9$

   Step-by-step solution

   1. $$u\cdot v=\sum _i{u}_i{v}_i$$
   2. $$u\cdot v=9$$
4. 4.

   Find the dot product.

   $$u=\left[\begin{array}{c}3\\ 0\\ -1\end{array}\right],v=\left[\begin{array}{c}-2\\ 5\\ 4\end{array}\right],u\cdot v=?$$

   Answer: $-10$

   Step-by-step solution

   1. $$u\cdot v=\sum _i{u}_i{v}_i$$
   2. $$u\cdot v=-10$$
5. 5.

   Find the dot product.

   $$u=\left[\begin{array}{c}1\\ 4\\ 2\end{array}\right],v=\left[\begin{array}{c}3\\ -1\\ 2\end{array}\right],u\cdot v=?$$

   Answer: $3$

   Step-by-step solution

   1. $$u\cdot v=\sum _i{u}_i{v}_i$$
   2. $$u\cdot v=3$$
6. 6.

   Find the dot product.

   $$u=\left[\begin{array}{c}-2\\ 3\\ 5\end{array}\right],v=\left[\begin{array}{c}4\\ 1\\ -2\end{array}\right],u\cdot v=?$$

   Answer: $-15$

   Step-by-step solution

   1. $$u\cdot v=\sum _i{u}_i{v}_i$$
   2. $$u\cdot v=-15$$
7. 7.

   Find the dot product.

   $$u=\left[\begin{array}{c}2\\ 3\end{array}\right],v=\left[\begin{array}{c}4\\ 1\end{array}\right],u\cdot v=?$$

   Answer: $11$

   Step-by-step solution

   1. $$u\cdot v=\sum _i{u}_i{v}_i$$
   2. $$u\cdot v=11$$
8. 8.

   Find the dot product.

   $$u=\left[\begin{array}{c}5\\ -2\end{array}\right],v=\left[\begin{array}{c}1\\ 3\end{array}\right],u\cdot v=?$$

   Answer: $-1$

   Step-by-step solution

   1. $$u\cdot v=\sum _i{u}_i{v}_i$$
   2. $$u\cdot v=-1$$
9. 9.

   Find the dot product.

   $$u=\left[\begin{array}{c}1\\ 1\\ 1\end{array}\right],v=\left[\begin{array}{c}2\\ 3\\ 4\end{array}\right],u\cdot v=?$$

   Answer: $9$

   Step-by-step solution

   1. $$u\cdot v=\sum _i{u}_i{v}_i$$
   2. $$u\cdot v=9$$
10. 10.

    Find the dot product.

    $$u=\left[\begin{array}{c}4\\ 0\\ 2\end{array}\right],v=\left[\begin{array}{c}1\\ 5\\ -3\end{array}\right],u\cdot v=?$$

    Answer: $-2$

    Step-by-step solution

    1. $$u\cdot v=\sum _i{u}_i{v}_i$$
    2. $$u\cdot v=-2$$
11. 11.

    Find the dot product.

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

    Answer: $5$

    Step-by-step solution

    1. $$u\cdot v=\sum _i{u}_i{v}_i$$
    2. $$u\cdot v=5$$
12. 12.

    Find the dot product.

    $$u=\left[\begin{array}{c}3\\ 6\end{array}\right],v=\left[\begin{array}{c}-2\\ 4\end{array}\right],u\cdot v=?$$

    Answer: $18$

    Step-by-step solution

    1. $$u\cdot v=\sum _i{u}_i{v}_i$$
    2. $$u\cdot v=18$$
13. 13.

    Find the dot product.

    $$u=\left[\begin{array}{c}2\\ -5\\ 1\end{array}\right],v=\left[\begin{array}{c}3\\ 1\\ 4\end{array}\right],u\cdot v=?$$

    Answer: $5$

    Step-by-step solution

    1. $$u\cdot v=\sum _i{u}_i{v}_i$$
    2. $$u\cdot v=5$$
14. 14.

    Find the dot product.

    $$u=\left[\begin{array}{c}7\\ 2\end{array}\right],v=\left[\begin{array}{c}1\\ -1\end{array}\right],u\cdot v=?$$

    Answer: $5$

    Step-by-step solution

    1. $$u\cdot v=\sum _i{u}_i{v}_i$$
    2. $$u\cdot v=5$$
15. 15.

    Find the dot product.

    $$u=\left[\begin{array}{c}0\\ 3\\ 5\end{array}\right],v=\left[\begin{array}{c}2\\ 1\\ -2\end{array}\right],u\cdot v=?$$

    Answer: $-7$

    Step-by-step solution

    1. $$u\cdot v=\sum _i{u}_i{v}_i$$
    2. $$u\cdot v=-7$$
16. 16.

    Find the dot product.

    $$u=\left[\begin{array}{c}4\\ -3\end{array}\right],v=\left[\begin{array}{c}2\\ 6\end{array}\right],u\cdot v=?$$

    Answer: $-10$

    Step-by-step solution

    1. $$u\cdot v=\sum _i{u}_i{v}_i$$
    2. $$u\cdot v=-10$$
17. 17.

    Find the dot product.

    $$u=\left[\begin{array}{c}1\\ 2\\ -4\end{array}\right],v=\left[\begin{array}{c}5\\ 0\\ 3\end{array}\right],u\cdot v=?$$

    Answer: $-7$

    Step-by-step solution

    1. $$u\cdot v=\sum _i{u}_i{v}_i$$
    2. $$u\cdot v=-7$$

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