# Angle Between Two Zero-Based Vectors in XY-Plane

**Problem**

What is the angle between zero-based vectors ${\bf V_1} = (-\sqrt{3}, ~ 1)$ and ${\bf V_2} = (2\sqrt{3}, ~ 2)$ in an *x*-*y* coordinate system?

A. 0° | C. 150° |

B. 180° | D. 120° |

**Answer Key**

[ D ]

**Solution**

Zero-based vector means origin-based vector

$\cos \theta = \dfrac{{\bf V_1} \cdot {\bf V_2}}{V_1 \cdot V_2}$

$\cos \theta = \dfrac{-\sqrt{3} \left( 2\sqrt{3} \right) + 1(2)}{\sqrt{\left( -\sqrt{3} \right)^2 + 1^2} \cdot \sqrt{\left( 2\sqrt{3} \right)^2 + 2^2}}$

$\cos \theta = -\frac{1}{2}$

$\theta = 120^\circ$ ← Answer: [ D ]

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