Find the normal line to the plan Ax+By+Cz+d=0 that passes through the point x_0,y_0,z_0

Find the scalar and vector projections of b onto a.
$a=(4,7,-4),b=(3,-1,1)$

Which of the following expressions are meaningful? Which are meaningless? Explain.
a) $(a\cdot b)\cdot c$
$(a\cdot b)\cdot c$ has ? because it is the dot product of ?.
b) $(a\cdot b)c$
$(a\cdot b)c$ has ? because it is a scalar multiple of ?.
c) $|a|(b\cdot c)$
$|a|(b\cdot c)$ has ? because it is the product of ?.
d) $a\cdot (b+c)$
$a\cdot (b+c)$ has ? because it is the dot product of ?.
e) $a\cdot b+c$
$a\cdot b+c$ has ? because it is the sum of ?.
f) $|a|\cdot (b+c)$
$|a|\cdot (b+c)$ has ? because it is the dot product of ?.

Let F be a fixed 3x2 matrix, and let H be the set of all matrices A in ${\displaystyle M_{2\times 4}}$ with the property that FA = 0 (the zero matrix in ${\displaystyle M_{3\times 4})}$. Determine if H is a subspace of ${\displaystyle M_{2\times 4}}$

Find a vector equation and parametric equations for the line segment that joins P to Q.
P(0, - 1, 1), Q(1/2, 1/3, 1/4)

I am stuck with the simple expression
${\displaystyle \frac{{\cos }^2(\theta +\alpha )}{1-{\cos }^2(\theta -\alpha )}=\text{const.}}$
where ${\displaystyle \theta }$ is a variable and ${\displaystyle \alpha }$ is the number satisfying
${\displaystyle \alpha ={\tan }^{-1}\left(\frac{4}{3}\right)}$

Prove that ${\displaystyle (1+2\cos \frac{2x}{3})\sin \frac{x}{3}=\sin x}$ for ${\displaystyle x\in \mathbb{R}}$

Calculate Password Possibilities With Specific Character Pattern
So i need to estimate the number of possibilities of a 8 characters password all stick together with a specified pattern wish have only:
uppercase letter and numbers. no special characters
The pattern is : Number, Number, Up Letter, Up Letter, Number, Up Letter, Number, Up Letter