If A and B have position vectors (1,2,3) and (4,5,6) respectively, find
(a) The direction vector of the line through A and B,
(b) The vector equation of the line through A and B

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.

If A and B have position vectors (1,2,3) and (4,5,6) respectively, find (a) The direction vector of the line through A and B, (b) The vector equation of the line through A and B

Learn how to calculate the direction vector and vector equation of a line passing through points A(1,2,3) and B(4,5,6). This problem covers key concepts in vector algebra and 3D analytic geometry. Suitable for students in Grades 10-12.

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