PART B:
To find the trigonometric form, we need the magnitude and angle of each vector.
The magnitude can be found with the expressions below:
[tex]\begin{gathered} ||u||=\sqrt{a^2+b^2}=\sqrt{5^2+(-11)^2}=\sqrt{25+121}=\sqrt{146}=12.08\\ \\ ||v||=\sqrt{a^2+b^2}=\sqrt{(-17)^2+9^2}=\sqrt{289+81}=\sqrt{370}=19.24 \end{gathered}[/tex]
And the angle of each vector can be found with the expressions below:
[tex]\begin{gathered} \theta_u=\tan^{-1}(\frac{b}{a})=\tan^{-1}(\frac{-11}{5})=294.44°\\ \\ \theta_v=\tan^{-1}(\frac{b}{a})=\tan^{-1}(\frac{9}{-17})=152.10° \end{gathered}[/tex]
Therefore vectors u and v in the trigonometric form are:
[tex]\begin{gathered} u=12.08(\cos294.44°i+\sin294.44°j)\\ \\ v=19.24(\cos152.10°i+\sin152.10°j) \end{gathered}[/tex]
PART C:
To find 7u - 4v, let's use the linear form of each vector, multiply by the constant values and subtract the results:
[tex]\begin{gathered} u=5i-11j\\ \\ 7u=7(5i-11j)=35i-77j\\ \\ v=-17i+9j\\ \\ 4v=4(-17i+9j)=-68i+36j\\ \\ 7u-4v=35i-77j-(-68i+36j)=103i-113j \end{gathered}[/tex]
