Find each dot product. Then state whether the angle between the vectors is \( >\mathbf{9 0 ^ { \circ }},\left\langle\mathbf{9 0 ^ { \circ }}\right. \), or \( =\mathbf{9 0 ^ { \circ }} \). \( \begin{array}{ll}\text { 7. }\langle 2,3\rangle \cdot\langle 5,7\rangle & \text { 8. }\langle-1,-4\rangle \cdot\langle 2,-3\rangle \\ \text { 9. }\langle 4,8\rangle \cdot\langle 4,-2\rangle & \text { 10. }\langle-4,6\rangle \cdot\langle 6,-9\rangle\end{array} \)

Let's calculate the dot products for each pair of vectors and analyze the angles. 1. For \(\langle 2, 3 \rangle \cdot \langle 5, 7 \rangle\): \[ \langle 2, 3 \rangle \cdot \langle 5, 7 \rangle = (2 \times 5) + (3 \times 7) = 10 + 21 = 31 \] To find the angle: \[ \text{Since } \text{dot product} > 0, \text{the angle } < 90^\circ. \] 2. For \(\langle -1, -4 \rangle \cdot \langle 2, -3 \rangle\): \[ \langle -1, -4 \rangle \cdot \langle 2, -3 \rangle = (-1 \times 2) + (-4 \times -3) = -2 + 12 = 10 \] Here, the dot product is also positive: \[ \text{Since } \text{dot product} > 0, \text{the angle } < 90^\circ. \] 3. For \(\langle 4, 8 \rangle \cdot \langle 4, -2 \rangle\): \[ \langle 4, 8 \rangle \cdot \langle 4, -2 \rangle = (4 \times 4) + (8 \times -2) = 16 - 16 = 0 \] A dot product of zero indicates: \[ \text{Angle } = 90^\circ. \] 4. For \(\langle -4, 6 \rangle \cdot \langle 6, -9 \rangle\): \[ \langle -4, 6 \rangle \cdot \langle 6, -9 \rangle = (-4 \times 6) + (6 \times -9) = -24 - 54 = -78 \] Here, the dot product is negative: \[ \text{Since } \text{dot product} < 0, \text{the angle } > 90^\circ. \] In summary: 7. Angle \( < 90^\circ \) with dot product of 31. 8. Angle \( < 90^\circ \) with dot product of 10. 9. Angle \( = 90^\circ \) with dot product of 0. 10. Angle \( > 90^\circ \) with dot product of -78.