Given two lines with direction vectors $(3,4)$ and $(5,12)$, use the Law of Cosines to find the acute angle between the lines.

• A. ${\cos }^{-1}\left(\frac{63}{65}\right)$
• B. ${\cos }^{-1}\left(\frac{24}{65}\right)$
• C. ${\cos }^{-1}\left(\frac{39}{60}\right)$
• D. ${\cos }^{-1}\left(\frac{56}{65}\right)$

Answer: (A)

Verified step by step guidance

1. Identify the given direction vectors of the two lines: \( \mathbf{v_1} = (3, 4) \) and \( \mathbf{v_2} = (5, 12) \).
2. Recall that the angle \( \theta \) between two vectors can be found using the dot product formula: \( \mathbf{v_1} \cdot \mathbf{v_2} = \|\mathbf{v_1}\| \|\mathbf{v_2}\| \cos \theta \).
3. Calculate the dot product of the two vectors: \( \mathbf{v_1} \cdot \mathbf{v_2} = 3 \times 5 + 4 \times 12 \).
4. Find the magnitudes (lengths) of each vector using the formula \( \|\mathbf{v}\| = \sqrt{x^2 + y^2} \): \( \|\mathbf{v_1}\| = \sqrt{3^2 + 4^2} \) and \( \|\mathbf{v_2}\| = \sqrt{5^2 + 12^2} \).
5. Use the Law of Cosines relationship from the dot product to solve for the angle: \[ \cos \theta = \frac{\mathbf{v_1} \cdot \mathbf{v_2}}{\|\mathbf{v_1}\| \|\mathbf{v_2}\|} \] then find \( \theta = \cos^{-1} \left( \frac{\mathbf{v_1} \cdot \mathbf{v_2}}{\|\mathbf{v_1}\| \|\mathbf{v_2}\|} \right) \).