Given the position vectors of points $A$ and $B$ as $\mathbf{A} = (1, 2, 3)$ and $\mathbf{B} = (4, 5, 6)$, we can solve the following:

(a) Direction Vector of the Line through A and B

The direction vector $\mathbf{d}$ of the line through points $A$ and $B$ is found by subtracting the position vector of $A$ from the position vector of $B$:

$\mathbf{d} = \mathbf{B} - \mathbf{A} = (4, 5, 6) - (1, 2, 3)$

$\mathbf{d} = (4 - 1, 5 - 2, 6 - 3) = (3, 3, 3)$

So, the direction vector $\mathbf{d}$ is $(3, 3, 3)$.

(b) Vector Equation of the Line through A and B

The vector equation of a line passing through a point $\mathbf{A} = (1, 2, 3)$ with direction vector $\mathbf{d} = (3, 3, 3)$ is given by:

$\mathbf{r}(t) = \mathbf{A} + t\mathbf{d}$

Substituting the values:

$\mathbf{r}(t) = (1, 2, 3) + t(3, 3, 3)$

Expanding this, we get:

$\mathbf{r}(t) = (1 + 3t, 2 + 3t, 3 + 3t)$

Thus, the vector equation of the line through $A$ and $B$ is:

$\mathbf{r}(t) = (1 + 3t, 2 + 3t, 3 + 3t)$

Would you like more details or have any questions? Here are some related questions to consider:

1. How do you find the magnitude of the direction vector $\mathbf{d}$?
2. Can you express the vector equation of the line in parametric form?
3. How do you determine if another point lies on the line through $A$ and $B$?
4. What would the line equation look like if $A$ and $B$ were in two dimensions instead of three?
5. How can you find the angle between two lines given their direction vectors?

Tip: When given two points, the direction vector is simply the difference between their coordinates.