Cosine Similarity
To determine if two vectors are similar using the formula you provided, we need to calculate the cosine of the angle between them. The formula for the cosine of the angle $\theta$ between two vectors $\mathbf{A}$ and $\mathbf{B}$ is given by:
\[cos(\theta) = \frac{\mathbf{A}\cdot\mathbf{B}}{||\mathbf{A}||\cdot||\mathbf{B}||}\]where $\mathbf{A}\cdot\mathbf{B}$ represents the dot product of the two vectors and $||\mathbf{A}||$ and $||\mathbf{B}||$ represent the magnitudes (or norms) of the two vectors.
In this case, we are given the vectors $\mathbf{A}=[50,10]$ and $\mathbf{B}=[14,12]$. To calculate their similarity, we first need to find their dot product:
\[\mathbf{A}\cdot\mathbf{B} = 50 \times 14 + 10 \times 12 = 700 + 120 = 820\]Next, we calculate the magnitudes of $\mathbf{A}$ and $\mathbf{B}$:
\[\|\|\mathbf{A}\|\| = \sqrt{50^2 + 10^2} = \sqrt{2500 + 100} = \sqrt{2600} \approx 50.99\] \[\|\|\mathbf{B}\|\| = \sqrt{14^2 + 12^2} = \sqrt{196 + 144} = \sqrt{340} \approx 18.73\]Now, we can use the formula to find the cosine of the angle between $\mathbf{A}$ and $\mathbf{B}$:
\[cos(\theta) = \frac{\mathbf{A}\cdot\mathbf{B}}{||\mathbf{A}||\cdot||\mathbf{B}||} = \frac{820}{\sqrt{2600 \times 340}} \approx 0.915\]Since $cos(\theta) \approx 0.915$, the vectors $\mathbf{A}$ and $\mathbf{B}$ are similar, meaning they have a small angle between them (approximately 18 degrees).