### Part (a)
Given the vector [tex]\( x = (1, 0, s, 2) \)[/tex] and the system of equations:
[tex]\[
\begin{array}{rcl}
4x_1 + 2x_2 - x_3 + 2x_4 & = & 0 \\
-3x_2 + x_3 + t x_4 & = & 0
\end{array}
\][/tex]
We need to determine the values of [tex]\( s \)[/tex] and [tex]\( t \)[/tex].
First, let's consider the first equation:
[tex]\[
4x_1 + 2x_2 - x_3 + 2x_4 = 0
\][/tex]
Substituting the values [tex]\( x_1 = 1 \)[/tex], [tex]\( x_2 = 0 \)[/tex], and [tex]\( x_4 = 2 \)[/tex], we get:
[tex]\[
4(1) + 2(0) - x_3 + 2(2) = 0
\][/tex]
Simplify the equation:
[tex]\[
4 + 0 - x_3 + 4 = 0
\][/tex]
[tex]\[
8 - x_3 = 0
\][/tex]
Solving for [tex]\( x_3 \)[/tex]:
[tex]\[
x_3 = 8
\][/tex]
So, [tex]\( s = 8 \)[/tex].
Now, let's consider the second equation:
[tex]\[
-3x_2 + x_3 + t x_4 = 0
\][/tex]
Substituting the values [tex]\( x_2 = 0 \)[/tex], [tex]\( x_3 = 8 \)[/tex], and [tex]\( x_4 = 2 \)[/tex], we get:
[tex]\[
-3(0) + 8 + t(2) = 0
\][/tex]
Simplify the equation:
[tex]\[
0 + 8 + 2t = 0
\][/tex]
[tex]\[
8 + 2t = 0
\][/tex]
Solving for [tex]\( t \)[/tex]:
[tex]\[
2t = -8
\][/tex]
[tex]\[
t = -4
\][/tex]
So, the values are [tex]\( s = 8 \)[/tex] and [tex]\( t = -4 \)[/tex].
### Part (b)
Given vectors [tex]\( u = (1, 2, 3) \)[/tex], [tex]\( v = (-3, -2, -1) \)[/tex], and [tex]\( w = (w_1, w_2, w_3) \)[/tex], we need to verify that:
[tex]\[
u \cdot (v + w) = u \cdot v + u \cdot w
\][/tex]
First, compute [tex]\( v + w \)[/tex]:
[tex]\[
v + w = (-3, -2, -1) + (w_1, w_2, w_3) = (-3 + w_1, -2 + w_2, -1 + w_3)
\][/tex]
Next, compute the dot product [tex]\( u \cdot (v + w) \)[/tex]:
[tex]\[
u \cdot (v + w) = (1, 2, 3) \cdot (-3 + w_1, -2 + w_2, -1 + w_3)
\][/tex]
[tex]\[
= 1(-3 + w_1) + 2(-2 + w_2) + 3(-1 + w_3)
\][/tex]
[tex]\[
= -3 + w_1 - 4 + 2w_2 - 3 + 3w_3
\][/tex]
[tex]\[
= w_1 + 2w_2 + 3w_3 - 10
\][/tex]
Now, compute the dot products [tex]\( u \cdot v \)[/tex] and [tex]\( u \cdot w \)[/tex]:
[tex]\[
u \cdot v = (1, 2, 3) \cdot (-3, -2, -1)
\][/tex]
[tex]\[
= 1(-3) + 2(-2) + 3(-1)
\][/tex]
[tex]\[
= -3 - 4 - 3
\][/tex]
[tex]\[
= -10
\][/tex]
[tex]\[
u \cdot w = (1, 2, 3) \cdot (w_1, w_2, w_3)
\][/tex]
[tex]\[
= 1w_1 + 2w_2 + 3w_3
\][/tex]
[tex]\[
= w_1 + 2w_2 + 3w_3
\][/tex]
Finally, verify that:
[tex]\[
u \cdot (v + w) = u \cdot v + u \cdot w
\][/tex]
[tex]\[
w_1 + 2w_2 + 3w_3 - 10 = -10 + (w_1 + 2w_2 + 3w_3)
\][/tex]
[tex]\[
w_1 + 2w_2 + 3w_3 - 10 = w_1 + 2w_2 + 3w_3 - 10
\][/tex]
Both sides of the equation are equal, verifying that:
[tex]\[
u \cdot (v + w) = u \cdot v + u \cdot w
\][/tex]
Thus, the properties hold true as expected.
