Vector $A$ has a magnitude of 10 units and makes an angle of $30^{\circ}$ with the positive $X$-axis. Vector $B$ has a magnitude of 20 units and makes an angle of $30^{\circ}$ with the negative $X$-axis. What is the magnitude of the resultant between these two vectors?

• A

  $20 \sqrt{3}$
• B

  $35$
• C

  $15 \sqrt{3}$
• Correct answer
  D

  $10 \sqrt{3}$

Solution

Given, $$ \begin{gathered} A=10 \text { units } \\ B=20 \text { units } \\ \theta=30^{\circ} \end{gathered} $$

The given situation is shown in the figure. The angle between $\mathbf{A}$ and $\mathbf{B}$, $$ \theta=180^{\circ}-30^{\circ}-30^{\circ}=120^{\circ} $$ $\therefore$ Magnitude of the resultant vector, $$ \begin{aligned} R &=\sqrt{A^{2}+B^{2}+2 A B \cos \theta} \\ &=\sqrt{10^{2}+20^{2}+2 \times 10 \times 20 \cos 120^{\circ}} \\ &=\sqrt{100+400-200} \\ &=10 \sqrt{3} \end{aligned} $$

Hence, the correct answer is option: 'D'.