The correct answer is the 3rd option.
The component form of a vector is to write in the form (x, y) the x and y component of a vector.
Since the vector starts at (0, 0) and ends at (5, -8) We can rest the start point to the end point to get the components of the vector:
[tex]\begin{gathered} x\text{ component:} \\ 5-0=5 \\ y\text{ component}\colon \\ -8-0=-8 \end{gathered}[/tex]
The component for is:
[tex]\langle5,-8\rangle[/tex]
The magnitude of a vector is its' module, we calculate it by:
[tex]\mleft\Vert u\mleft\Vert=\sqrt[]{x^2+y^2}\mright?\mright?[/tex]
Where x and y are the components of the vector. In this case:
[tex]\mleft\Vert u\mleft\Vert=\sqrt[]{5^2+(-8)^2}=\sqrt[]{25+64}=\sqrt[]{89}\mright?\mright?[/tex]
The whole answer is:
[tex]u=\langle5,-8\rangle;\mleft\Vert u\mleft\Vert=\sqrt[]{89}\mright?\mright?[/tex]
Which is option 3.
