Question

Could someone help me find the magnitude and degrees of the vectors with the given information.

12i + 4sqrt3j

Answer 1

`\bf \stackrel{a}{12}i+\stackrel{b}{4√(3)}j\implies \ \textless \ 12~,~4√(3)\ \textgreater \ \qquad \begin{cases} \stackrel{magnitude}{r}=√(a^2+b^2)\\\\ \theta =tan^(-1)\left( (b)/(a) \right) \end{cases}\\\\ -------------------------------\\\\ r=\sqrt{12^2+(4√(3))^2}\implies r=\sqrt{144+(4^2√(3^2))}`


`\bf r=√(144+(16\cdot 3))\implies r=√(144+48)\implies r=√(192) \\\\\\ r=√(64\cdot 3)\implies r=√(8^2\cdot 3)\implies r=8√(3)\\\\ -------------------------------\\\\ \theta =tan^(-1)\left( \cfrac{4√(3)}{12} \right)\implies \theta =tan^(-1)\left( \cfrac{√(3)}{3} \right)\implies \theta = \begin{cases} (\pi )/(6)\leftarrow \\\\ (7\pi )/(6) \end{cases}`

notice, those two angles have a valid tangent value, however, notice the "a" and "b" components, their signs are +a and +b, meaning the angle is in the first quadrant, thus, it has to be angle in the first quadrant then.