Calculate, correct to the nearest degree, the angle between the vectors \(\begin{pmatrix} 13 \\ 1 \end{pmatrix}\) and \(\begin{pmatrix} 1 \\ 4 \end{pmatrix}...

Calculate, correct to the nearest degree, the angle between the vectors \(\begin{pmatrix} 13 \\ 1 \end{pmatrix}\) and \(\begin{pmatrix} 1 \\ 4 \end{pmatrix}\).

Answer Details

To find the angle between two vectors, we can use the dot product formula: \(\mathbf{a} \cdot \mathbf{b} = \|\mathbf{a}\| \|\mathbf{b}\| \cos \theta\), where \(\mathbf{a}\) and \(\mathbf{b}\) are the vectors, \(\|\mathbf{a}\|\) and \(\|\mathbf{b}\|\) are their magnitudes, and \(\theta\) is the angle between them. So, for the given vectors, we have: \(\begin{pmatrix} 13 \\ 1 \end{pmatrix} \cdot \begin{pmatrix} 1 \\ 4 \end{pmatrix} = \sqrt{13^2 + 1^2} \sqrt{1^2 + 4^2} \cos \theta\) Simplifying, we get: \(53 = \sqrt{170} \sqrt{17} \cos \theta\) \(\cos \theta = \frac{53}{\sqrt{170} \sqrt{17}}\) Taking the inverse cosine of both sides, we get: \(\theta \approx 73^\circ\) So, the answer is closest to 72°.