Given vectors v1 = i + 4j + 2k, v2 = 3i − 3j − 2k, and v3 = -2i + 2j + 6k, find the magnitude of their sum.

Let the three vectors be $\vec{v_1}$, $\vec{v_2}$, and $\vec{v_3}$.

$$\vec{v_1} = \hat{i} + 4\hat{j} + 2\hat{k}$$ $$\vec{v_2} = 3\hat{i} - 3\hat{j} - 2\hat{k}$$ $$\vec{v_3} = -2\hat{i} + 2\hat{j} + 6\hat{k}$$

Let $\vec{S}$ be the sum of the vectors.

$$\vec{S} = \vec{v_1} + \vec{v_2} + \vec{v_3}$$ $$\vec{S} = (\hat{i} + 4\hat{j} + 2\hat{k}) + (3\hat{i} - 3\hat{j} - 2\hat{k}) + (-2\hat{i} + 2\hat{j} + 6\hat{k})$$

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

$$\vec{S} = (1+3-2)\hat{i} + (4-3+2)\hat{j} + (2-2+6)\hat{k}$$ $$\vec{S} = 2\hat{i} + 3\hat{j} + 6\hat{k}$$

The modulus of a vector $x\hat{i} + y\hat{j} + z\hat{k}$ is given by $\sqrt{x^2 + y^2 + z^2}$.

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

The calculated modulus is 7. This matches option (b) for question 8.

The final answer is $\text{(b)} 7$.