Find the value of m for which the vectors -2mi + 2j - 2k and 3i + 4mj + 3k perpendicular to each other.

Let $\vec{A} = -2m\vec{i} + 2\vec{j} - 2\vec{k}$ and $\vec{B} = 3\vec{i} + 4m\vec{j} + 3\vec{k}$.

Two vectors are perpendicular if their dot product is zero ($\vec{A} \cdot \vec{B} = 0$).

The value of $m$ is $\boxed{3}$.

The resultant vector is the sum of $\vec{a}_1$ and $\vec{a}_2$.

The unit vector $\hat{u}$ is $\frac{\vec{R}}{|\vec{R}|}$.

The unit vector parallel to the resultant is $\boxed*{\frac{9}{\sqrt{85}}\vec{i} - \frac{2}{\sqrt{85}}\vec{j}*}$.

Let $\vec{A} = 3\vec{i} - 4\vec{j} + 12\vec{k}$ and $\vec{B} = \vec{i} + 2\vec{j} - \vec{k}$.

The unit vector $\hat{u}$ is $\frac{\vec{C}}{|\vec{C}|}$.

The unit vector perpendicular to the given vectors is $\boxed*{-\frac{4}{\sqrt{29}}\vec{i} + \frac{3}{\sqrt{29}}\vec{j} + \frac{2}{\sqrt{29}}\vec{k}*}$.

i. Magnitude of a vector
The magnitude of a vector is its length or size, representing the scalar value of the vector quantity. For a vector $\vec{v} = x\vec{i} + y\vec{j} + z\vec{k}$, its magnitude is given by $|\vec{v}| = \sqrt{x^2 + y^2 + z^2}$.

ii. Directional Cosines
Directional cosines are the cosines of the angles that a vector makes with the positive x, y, and z axes. For a vector $\vec{v} = x\vec{i} + y\vec{j} + z\vec{k}$, the directional cosines are $\cos\alpha = \frac{x}{|\vec{v}|}$, $\cos\beta = \frac{y}{|\vec{v}|}$, and $\cos\gamma = \frac{z}{|\vec{v}|}$.

iii. Parallelism of two vectors
Two vectors are parallel if they point in the same or opposite direction. Mathematically, two non-zero vectors $\vec{A}$ and $\vec{B}$ are parallel if one is a scalar multiple of the other, i.e., $\vec{A} = k\vec{B}$ for some scalar $k$.