E and F are vector fields given by E=2xa_x +a_y +yza_z and F=xya_x -y^2 a_y +xyza_z. Determine (a)|E| at (1,2,3)

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 ?.

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)

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

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 ${\displaystyle M_{2\times 4}}$

Let's say I have 3 variables (X, Y and Z) each one can assume a positive integer value, including zero (0, 1, 2, ..., 100), but the sum of the three must be 100. How many combinations do I have ? Different variables can have the same value.

I am trying to prove that, given two linearly independent vectors $v_1,v_2$, the sum of vectors $v_3+v_4$ (both being some linear combination of $v_1,v_2$) is linearly dependent with $v_1,v_2$
It makes sense to me intuitively since if we have $x_1{v}_1+x_2{v}_2=v_3$ and $y_1{v}_1+y_2{v}_2=v_4$, then $v_3+v_4=x_1{v}_1+x_2{v}_2+y_1{v}_1+y_2{v}_2$, but I'm struggling to formulate a proof using the definition of linear dependency. Any suggestions on how I might do so?

I'm trying to integrate:
${\displaystyle \int \frac{8dx}{3\cos 2x+1}}$