Question

Vector u has its initial point at (-7, 2) and its terminal point at (11, -5). Vector v has a direction opposite that of vector u, and its magnitude is three times the magnitude of u. What is the component form of vector v?

Answer 1

$\bf u\implies \begin{cases} (-7,2)\\ (11,5) \end{cases}\implies [11-(-7)]~,~[5-2]\implies (11+7)~,~(5-2) \\\\\\ \stackrel{\textit{component form}}{\ \textless \ 18~,~3\ \textgreater \ }\\\\ -------------------------------\\\\ ||u||=\sqrt{18^2+3^2}\implies ||u||=\sqrt{333}\implies ||u||=\sqrt{9\cdot 37} \\\\\\ ||u||=\sqrt{3^2\cdot 37}\implies ||u||=3\sqrt{37}\\\\ -------------------------------$

$\bf \ \textless \ -18~,~-3\ \textgreater \ \impliedby \textit{opposite vector to \underline{u}} \\\\\\ 3\cdot 3\sqrt{37}\implies 9\sqrt{37}\impliedby \textit{3 times the magnitude of \underline{u}} \\\\\\ 3\ \textless \ -18,-3\ \textgreater \ \implies \stackrel{\textit{component form of "v"}}{\ \textless \ -54~,~-9\ \textgreater \ }\impliedby \textit{3 times as long as \underline{u}}$

Answer 2

Answer with explanation:

Initial Point of vector , u= (-7,2)= -7 i +  2 j

Terminal point of Vector , u= (11, -5)= 11 i -5 j

Vector , u= Terminal point - Initial Point

                = 11 i - 5 j - (-7 i+  2 j)

                 = 11  i - 5 j + 7 i - 2 j

                 = 18 i - 7 j

⇒Vector, v= 3 ×Direction Opposite to that of vector, u

              = -3 u

             = -3 × (18 i - 7 j )

             = 3× (-18 i + 7 j)

              = -54  i + 21 j