Motion in a Plane

Frequency: $n = \dfrac{22}{44} = \dfrac{1}{2}$ revolution per second
Angular speed: $\omega = 2\pi n = 2\pi \times \dfrac{1}{2} = \pi$ rad/s
Centripetal acceleration: $a_r = \omega^2 r = (2\pi n)^2 r$
$a_r = (\pi)^2 \times 1 = \pi^2$ m s$^{-2}$
The speed is constant, so the tangential acceleration is zero: $a_T = 0$.
So the acceleration is $\pi^2$ m s$^{-2}$, directed along the radius towards the centre.

Two vectors are perpendicular when their scalar (dot) product is zero.
$(2\hat{i} + 3\hat{j} + 8\hat{k})\cdot(-4\hat{i} + 4\hat{j} + \alpha\hat{k}) = 0$
$2 \times (-4) + 3 \times 4 + 8 \times \alpha = 0$
$-8 + 12 + 8\alpha = 0$
$8\alpha = -4$
$\alpha = -\dfrac{1}{2}$

$\vec{B} \times \vec{A} = BA\sin\theta\,\hat{n}$, where $\hat{n}$ is a unit vector perpendicular to the plane containing $\vec{A}$ and $\vec{B}$.
$(\vec{B} \times \vec{A})\cdot\vec{A} = (BA\sin\theta)(\hat{n}\cdot\vec{A})$
Since $\hat{n}$ is perpendicular to $\vec{A}$, $\hat{n}\cdot\vec{A} = A\cos 90^\circ = 0$.
$\Rightarrow (\vec{B} \times \vec{A})\cdot\vec{A} = 0$
In general, a scalar triple product with two identical vectors is always zero.