Question

Write the vector v in terms of i and j whose magnitude and direction angle θ are given.

Answer 1

$\bf \begin{cases} r=8\\ \theta =30^o \end{cases}\qquad \qquad \qquad \qquad \begin{cases} tan(\theta )=\frac{b}{a}\\ r^2=a^2+b^2 \end{cases}\\\\ -------------------------------\\\\ tan(30^o)=\cfrac{b}{a}\implies \cfrac{\quad\frac{1}{2} \quad }{\frac{\sqrt{3}}{2}}=\cfrac{b}{a}\implies \cfrac{1}{\sqrt{3}}=\cfrac{b}{a}\implies \boxed{\cfrac{a}{\sqrt{3}}=b}\\\\ -------------------------------$

$\bf 8^2=a^2+b^2\implies 64=a^2+b^2\implies 64=a^2+\left( \boxed{\cfrac{a}{\sqrt{3}}} \right)^2 \\\\\\ 64=a^2+\cfrac{a^2}{3}\implies 64=\cfrac{4a^2}{3}\implies 16=\cfrac{a^2}{3}\implies 48=a^2 \\\\\\ \sqrt{48}=a\implies \boxed{4\sqrt{3}=a}\\\\ -------------------------------\\\\ \cfrac{a}{\sqrt{3}}=b\implies \cfrac{4\sqrt{3}}{\sqrt{3}}=b\implies \boxed{4=b}\\\\ -------------------------------\\\\ v~=~4\sqrt{3}i~+~4j$