P(3,2) Q(-1,5) and R(2,1) are vertices of a triangle. (i) Find the length of sides of the triangle. (ii) Use the dot product to find angles of the triangle.

$$RP = \sqrt{(3-2)^2 + (2-1)^2} = \sqrt{(1)^2 + (1)^2} = \sqrt{1 + 1} = \sqrt{2} \approx 1.414$$

The lengths of the sides are $\boxed{\mathbf{PQ=5, QR=5, RP=\sqrt{2} \approx 1.414}}$.

First, find the vectors $\vec{PQ}$ and $\vec{PR}$.

$$\vec{PQ} = Q - P = (-1-3, 5-2) = (-4, 3)$$ $$\vec{PR} = R - P = (2-3, 1-2) = (-1, -1)$$

Now, calculate their dot product and magnitudes:

$$\vec{PQ} \cdot \vec{PR} = (-4)(-1) + (3)(-1) = 4 - 3 = 1$$ $$|\vec{PQ}| = 5$$ $$|\vec{PR}| = \sqrt{(-1)^2 + (-1)^2} = \sqrt{1+1} = \sqrt{2}$$

Using the formula $\cos \theta = \frac{\vec{A} \cdot \vec{B}}{|\vec{A}| |\vec{B}|}$:

$$\cos P = \frac{1}{5 \cdot \sqrt{2}} = \frac{1}{5\sqrt{2}}$$ $$P = \arccos\left(\frac{1}{5\sqrt{2}}\right) \approx 81.87^\circ$$

First, find the vectors $\vec{QP}$ and $\vec{QR}$.

$$\vec{QP} = P - Q = (3-(-1), 2-5) = (4, -3)$$ $$\vec{QR} = R - Q = (2-(-1), 1-5) = (3, -4)$$

Now, calculate their dot product and magnitudes:

$$\vec{QP} \cdot \vec{QR} = (4)(3) + (-3)(-4) = 12 + 12 = 24$$ $$|\vec{QP}| = \sqrt{4^2 + (-3)^2} = \sqrt{16+9} = \sqrt{25} = 5$$ $$|\vec{QR}| = \sqrt{3^2 + (-4)^2} = \sqrt{9+16} = \sqrt{25} = 5$$

Using the formula $\cos \theta = \frac{\vec{A} \cdot \vec{B}}{|\vec{A}| |\vec{B}|}$:

$$\cos Q = \frac{24}{5 \cdot 5} = \frac{24}{25}$$ $$Q = \arccos\left(\frac{24}{25}\right) \approx 16.26^\circ$$

First, find the vectors $\vec{RP}$ and $\vec{RQ}$.

$$\vec{RP} = P - R = (3-2, 2-1) = (1, 1)$$ $$\vec{RQ} = Q - R = (-1-2, 5-1) = (-3, 4)$$

Now, calculate their dot product and magnitudes:

$$\vec{RP} \cdot \vec{RQ} = (1)(-3) + (1)(4) = -3 + 4 = 1$$ $$|\vec{RP}| = \sqrt{1^2 + 1^2} = \sqrt{2}$$ $$|\vec{RQ}| = \sqrt{(-3)^2 + 4^2} = \sqrt{9+16} = \sqrt{25} = 5$$

Using the formula $\cos \theta = \frac{\vec{A} \cdot \vec{B}}{|\vec{A}| |\vec{B}|}$:

$$\cos R = \frac{1}{\sqrt{2} \cdot 5} = \frac{1}{5\sqrt{2}}$$ $$R = \arccos\left(\frac{1}{5\sqrt{2}}\right) \approx 81.87^\circ$$

The angles of the triangle are $\boxed{\mathbf{P \approx 81.87^\circ, Q \approx 16.26^\circ, R \approx 81.87^\circ}}$.