Which of the following dopants is used in germanium to form an $n$-type semiconductor?

The figure shows $ABCDEF$ as a regular hexagon. What is the value of $\overrightarrow {AB} + \overrightarrow {AC} + \overrightarrow {AD} + \overrightarrow {AE} + \overrightarrow {AF}$ in terms of $\overrightarrow {AO}$?

If the vector $\vec{P} = 3\hat{i} + 4\hat{j} + 12\hat{k}$,then the magnitude of the vector $\vec{P}$ is ......

If $\vec{P} \cdot \vec{Q} = 0$ and $\vec{P} \times \vec{Q} = PQ$,then what is the angle between $\vec{P}$ and $\vec{Q}$ (in $^{\circ}$)?

State with reasons,whether the following algebraic operations with scalar and vector physical quantities are meaningful:
$(a)$ adding any two scalars,
$(b)$ adding a scalar to a vector of the same dimensions,
$(c)$ multiplying any vector by any scalar,
$(d)$ multiplying any two scalars,
$(e)$ adding any two vectors,
$(f)$ adding a component of a vector to the same vector.

Find the unit vector of a vector whose coordinates are $(8, 6)$.