Let's solve each of the problems step by step.

1.18 Find the vector with the initial point $P(4,3)$ and $Q(-9,-4)$ as the terminal point.

To find the vector $\overrightarrow{PQ}$, subtract the coordinates of the initial point $P$ from the coordinates of the terminal point $Q$: $\overrightarrow{PQ} = \langle Q_x - P_x, Q_y - P_y \rangle = \langle -9 - 4, -4 - 3 \rangle = \langle -13, -7 \rangle$ So, the vector is $\overrightarrow{PQ} = \langle -13, -7 \rangle$.

1.19 Given that $\mathbf{u} = \langle 10, -1 \rangle$ and $\mathbf{v} = \langle 10, -4 \rangle$, calculate $|\mathbf{u} - \mathbf{v}|$.

First, subtract the vectors $\mathbf{u}$ and $\mathbf{v}$: $\mathbf{u} - \mathbf{v} = \langle 10 - 10, -1 - (-4) \rangle = \langle 0, 3 \rangle$ Now, calculate the magnitude $|\mathbf{u} - \mathbf{v}|$: $|\mathbf{u} - \mathbf{v}| = \sqrt{0^2 + 3^2} = \sqrt{9} = 3$ So, $|\mathbf{u} - \mathbf{v}| = 3$.

1.20 Given that point $A$ has coordinates $(2,7)$ and $B$ has coordinates $(4,11)$, find the vector $\overrightarrow{BA}$.

To find the vector $\overrightarrow{BA}$, subtract the coordinates of $B$ from those of $A$: $\overrightarrow{BA} = \langle A_x - B_x, A_y - B_y \rangle = \langle 2 - 4, 7 - 11 \rangle = \langle -2, -4 \rangle$ So, the vector is $\overrightarrow{BA} = \langle -2, -4 \rangle$.

1.21 Perform the indicated operations and simplify $(c + 72)^2$.

Expand the expression: $(c + 72)^2 = c^2 + 2 \cdot 72 \cdot c + 72^2 = c^2 + 144c + 5184$ So, the simplified expression is $c^2 + 144c + 5184$.

1.22 Simplify the following expression: $5(3t - 4) - (t^2 + 1) - 4t(t - 1)$.

First, expand and distribute: $5(3t - 4) = 15t - 20$ $-4t(t - 1) = -4t^2 + 4t$ Now, combine all terms: $15t - 20 - (t^2 + 1) - 4t^2 + 4t = -5t^2 + 19t - 21$ So, the simplified expression is $-5t^2 + 19t - 21$.

1.23 Solve for $x$ in the equation $\cos(x) = -\frac{\sqrt{2}}{2}$.

The solutions for $\cos(x) = -\frac{\sqrt{2}}{2}$ are: $x = \frac{3\pi}{4} + 2k\pi \quad \text{or} \quad x = \frac{5\pi}{4} + 2k\pi \quad \text{for integer } k$

1.24 Convert $\log_8 4096 = 4$ to exponential form.

The equivalent exponential form is: $8^4 = 4096$

1.25 Given that $f(x) = x + 2$ and $g(x) = x^2$, find $(g \circ f)(x)$.

To find $g(f(x))$, substitute $f(x)$ into $g(x)$: $g(f(x)) = g(x + 2) = (x + 2)^2 = x^2 + 4x + 4$

1.26 Given the functions $f(x) = x^2 - 5x + 38$ and $g(x) = x - 17$, find $(f - g)(x)$.

Subtract $g(x)$ from $f(x)$: $(f - g)(x) = (x^2 - 5x + 38) - (x - 17) = x^2 - 6x + 55$

1.27 Use the table below to find $(f + g)(-1)$.

To find $(f + g)(-1)$, add $f(-1)$ and $g(-1)$: $f(-1) + g(-1) = 8 + (-7) = 1$

So, $(f + g)(-1) = 1$.

If you have any questions or need further details, feel free to ask!

Here are some related questions:

1. How do you find the magnitude of a vector?
2. What are the properties of the dot product of two vectors?
3. How do you solve trigonometric equations involving cosine?
4. What is the importance of converting logarithmic expressions to exponential form?
5. How do you perform function composition with more complex functions?

Tip: When solving vector problems, always double-check the direction and magnitude to ensure accuracy.