Find the vector of the line joining (1,2,3) and (-3, 4,3) and show that it is perpendicular to the z-axis.

Answer

To find the vector of the line joining the points \( A(1, 2, 3) \) and \( B(-3, 4, 3) \), and to show that it is perpendicular to the z-axis, we can follow these steps:

### Step 1: Find the direction vector of the line
The direction vector \( \vec{AB} \) from point \( A \) to point \( B \) can be calculated using the formula:
\[ \vec{AB} = \vec{B} - \vec{A} \]
Where \( \vec{A} = (1, 2, 3) \) and \( \vec{B} = (-3, 4, 3) \).

Calculating \( \vec{AB} \):
\[ \vec{AB} = (-3 - 1, 4 - 2, 3 - 3) = (-4, 2, 0) \]

### Step 2: Write the vector equation of the line
The vector equation of the line can be expressed as:
\[ \vec{r} = \vec{A} + \lambda \vec{AB} \]
Substituting \( \vec{A} = (1, 2, 3) \) and \( \vec{AB} = (-4, 2, 0) \):
\[ \vec{r} = (1, 2, 3) + \lambda (-4, 2, 0) \]

### Step 3: Show that the line is perpendicular to the z-axis
A line is perpendicular to the z-axis if its direction vector has no component in the z-direction. The z-component of the direction vector \( \vec{AB} = (-4, 2, 0) \) is \( 0 \).

Since the z-component is \( 0 \), this indicates that the line does not change in the z-direction as we move along it, confirming that it is indeed perpendicular to the z-axis.

### Conclusion
The vector of the line joining the points \( (1, 2, 3) \) and \( (-3, 4, 3) \) is \( (-4, 2, 0) \), and since the z-component is \( 0 \), the line is perpendicular to the z-axis.

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