Help
Well let's look at the very first one:
We are supposed to prove that:
$0[x_1,x_2,\dots,x_n] = [0,0,\dots,0]$.
By definition of the scalar product, for any $r \in \Bbb R$, we have:
$r[x_1,x_2,\dots,x_n] = [rx_1,rx_2,\dots,rx_n]$.
Letting $r = 0$ in the above leads to:
$0[x_1,x_2,\dots,x_n] = [0x_1,0x_2,\dots,0x_n]$.
Now, for any $j = 1,2,\dots,n$, is it true that $0x_j = 0$?
Ok, suppose $u = [x,y,z]$ (or we could write $u = [x_1,x_2,x_3]$ it's just notation).
By definition, $cu = c[x,y,z] = [cx,cy,cz]$.
If this equals $[0,0,0]$, this means:
$cx = 0$
$cy = 0$
$cz = 0$.
Now if $c = 0$, we have nothing to prove (see answer to number 1 above). So suppose $c \neq 0$. We need to show that in this case $x = y = z = 0$.
Since we are assuming $c \neq 0$, from the 3 equations above, we have:
$x = \dfrac{cx}{c} = \dfrac{0}{c} = 0$
and similarly for $y$ and $z$. Any questions?
3. Just set $\displaystyle \begin{align*} \mathbf{x} = \left( x_1, x_2, x_3 \right) , \mathbf{y} = \left( y_1, y_2, y_3 \right) , z = \left( z_1, z_2, z_3 \right) \end{align*}$ and see if the associative property holds.
4. A vector, say $\displaystyle \begin{align*} \mathbf{a} \end{align*}$, is a linear combination of two other vectors, say $\displaystyle \begin{align*} \mathbf{b} \end{align*}$ and $\displaystyle \begin{align*} \mathbf{c} \end{align*}$ if it's possible to write $\displaystyle \begin{align*} \mathbf{a} = c_1 \mathbf{b} + c_2\mathbf{c} \end{align*}$, where $\displaystyle \begin{align*} c_1, c_2 \end{align*}$ are constants.
5. A property of the dot product that will help you: $\displaystyle \begin{align*} \mathbf{a} \cdot \left( \mathbf{b} \cdot \mathbf{c} \right) = \mathbf{a} \cdot \mathbf{b} + \mathbf{a} \cdot \mathbf{c} \end{align*}$.
Also, you should suspect vector subtraction is NOT associative, since subtraction isn't even associative in the real numbers:
5-(4-3) = 5 - 1 = 4
(5-4)-3 = 1 - 3 = -2.
More abstractly:
a - (b - c) = a - b + c
(a - b) - c = a - b - c, and if these two are to be equal, we get c = -c, which isn't always the case.
We can write any point on the line L as (1+t,2t-1) for some t. Suppose n = (a,b).
Taking the dot product we have n.(1+t,2t-1) = a(1+t) + b(2t-1).
Let's use some specific values for t, to see what happens. At t = 0, we get the point on L, (1,-1). So if this solves n.x = c, for x = (1,-1), we have that c = a - b.
At t = 1, we get the point (2,1) on L, so plugging this into n.x = c gives us c = 2a + b. Since we are using the same "c", this means:
2a + b = a - b, or: a = -2b. So n is of the form (-2b,b).
Suppose we take b = 1. Then n = (-2,1), and c = -3.
Let's verify that this works:
(-2,1).(1+t,2t-1) = -2(1+t) + (2t-1) = -2 - 2t + 2t - 1 = -3.
Why this is called "normal form": note that L is parallel to the line L' = t(1,2), and that n.(1,2) = (-2,1).(1,2) = -2 + 2 = 0, in other words, n is perpendicular (normal) to L.
Let's "turn this problem on its head" by working backwards:
Suppose we have a NON-ZERO vector n = (a,b), and seek all points (x,y) that satisfy (a,b).(x,y) = c, that is:
ax + by = c, or equivalently, by = -ax + c.
We distinguish two cases:
Case 1: b = 0. In this case, then a is non-zero, so we get the (vertical) line:
x = c/a, which can be written as: (x,y) = (c/a,0) + t(0,1). Note that (a,0) and (0,1) are perpendicular.
Case 2: b is non-zero, so y = -(a/b)x + (c/b), which can be written as the line (x,y) = (0,c/b) + t(1,-(a/b)). Note here, as well: (a,b).(1,-(a/b)) = a - a = 0.
In this problem, our line L has a finite slope, so b is not 0.
Since we must have (1,2) parallel to (1,-(a/b)), this means a/b = -2.
Now (1,-1) lies on L, so writing (1,-1) = (0,c/b) + t(1,-(a/b)) = (t,(c-ta)/b)) = (t,(c+2bt)/b)), and solving for t, we see t = 1, so:
-1 = (c+2b)/b = (c/b) + 2, that is: c/b = -3.
Hence the normal form of L is any equation:
b(-2,1).(x,y) = -3b, and cancelling the b's, we get L has the equation: (-2,1).(x,y) = -3.
(Note the normal form isn't unique, we could multiply the constant c and the normal vector n by the same scalar).