Given vectors AB = 3i + 4j − 2k, BC = 4i + 3j + 5k, PQ = 2i + 3j + 6k, find the magnitude and the sum of each vector.

The magnitude of a vector $\vec{V} = x\mathbf{i} + y\mathbf{j} + z\mathbf{k}$ is given by $|\vec{V}| = \sqrt{x^2 + y^2 + z^2}$.
For $\vec{AB} = 3\mathbf{i} + 4\mathbf{j} - 2\mathbf{k}$:

$$|\vec{AB}| = \sqrt{3^2 + 4^2 + (-2)^2}$$ $$|\vec{AB}| = \sqrt{9 + 16 + 4}$$ $$|\vec{AB}| = \sqrt{29}$$

For $\vec{BC} = 4\mathbf{i} + 3\mathbf{j} + 5\mathbf{k}$:

$$|\vec{BC}| = \sqrt{4^2 + 3^2 + 5^2}$$ $$|\vec{BC}| = \sqrt{16 + 9 + 25}$$ $$|\vec{BC}| = \sqrt{50}$$ $$|\vec{BC}| = 5\sqrt{2}$$

For $\vec{PQ} = 2\mathbf{i} + 3\mathbf{j} + 6\mathbf{k}$:

$$|\vec{PQ}| = \sqrt{2^2 + 3^2 + 6^2}$$ $$|\vec{PQ}| = \sqrt{4 + 9 + 36}$$ $$|\vec{PQ}| = \sqrt{49}$$ $$|\vec{PQ}| = 7$$

Let $\vec{S}$ be the sum of the three vectors: $\vec{S} = \vec{AB} + \vec{BC} + \vec{PQ}$.

$$\vec{S} = (3\mathbf{i} + 4\mathbf{j} - 2\mathbf{k}) + (4\mathbf{i} + 3\mathbf{j} + 5\mathbf{k}) + (2\mathbf{i} + 3\mathbf{j} + 6\mathbf{k})$$

Combine the $\mathbf{i}$, $\mathbf{j}$, and $\mathbf{k}$ components:

$$\vec{S} = (3+4+2)\mathbf{i} + (4+3+3)\mathbf{j} + (-2+5+6)\mathbf{k}$$ $$\vec{S} = 9\mathbf{i} + 10\mathbf{j} + 9\mathbf{k}$$

For $\vec{S} = 9\mathbf{i} + 10\mathbf{j} + 9\mathbf{k}$:

$$|\vec{S}| = \sqrt{9^2 + 10^2 + 9^2}$$ $$|\vec{S}| = \sqrt{81 + 100 + 81}$$ $$|\vec{S}| = \sqrt{262}$$

The magnitudes of the individual vectors are:

• $|\vec{AB}| = \boxed{\sqrt{29}}$
• $|\vec{BC}| = \boxed{5\sqrt{2}}$
• $|\vec{PQ}| = \boxed{7}$

The sum of the three vectors is:

• $\vec{S} = \boxed{9\mathbf{i} + 10\mathbf{j} + 9\mathbf{k}}$

The magnitude of the sum of the three vectors is:

• $|\vec{S}| = \boxed{\sqrt{262}}$