Questions: Find the angle between the vectors. (First find an exact expression and then approximate to the nearest degree.) a=⟨2,3⟩, b=⟨4,-1⟩ exact approximate

Solution Steps

The dot product of the vectors \( \mathbf{a} = \langle 2, 3 \rangle \) and \( \mathbf{b} = \langle 4, -1 \rangle \) is calculated as follows: \[ \mathbf{a} \cdot \mathbf{b} = 2 \cdot 4 + 3 \cdot (-1) = 8 - 3 = 5 \]

The magnitudes of the vectors \( \mathbf{a} \) and \( \mathbf{b} \) are computed using the formula \( \|\mathbf{v}\| = \sqrt{x^2 + y^2} \): \[ \|\mathbf{a}\| = \sqrt{2^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13} \approx 3.6056 \] \[ \|\mathbf{b}\| = \sqrt{4^2 + (-1)^2} = \sqrt{16 + 1} = \sqrt{17} \approx 4.1231 \]

Using the dot product and the magnitudes, we find the cosine of the angle \( \theta \): \[ \cos(\theta) = \frac{\mathbf{a} \cdot \mathbf{b}}{\|\mathbf{a}\| \|\mathbf{b}\|} = \frac{5}{3.6056 \cdot 4.1231} \approx 0.3363 \]

To find the angle \( \theta \), we use the arccosine function: \[ \theta_{\text{radians}} = \arccos(0.3363) \approx 1.2278 \] Converting this to degrees: \[ \theta_{\text{degrees}} \approx 70.3462 \text{ degrees} \approx 70 \text{ degrees} \]

Final Answer