Step 1: Understanding the Concept:
Finding the distance between two points in 3D space is an extension of the Pythagorean theorem.
We calculate the difference between coordinates for each axis (x, y, and z), square them to ensure positive values, sum them, and finally take the square root of the total.
This gives the "Euclidean distance" or the length of the straight line segment connecting the points.
Step 2: Key Formula or Approach:
The distance \( d \) between points \( (x_1, y_1, z_1) \) and \( (x_2, y_2, z_2) \) is given by:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2 + (z_2 - z_1)^2} \]
Step 3: Detailed Explanation:
The given points are:
\( A(x_1, y_1, z_1) = (1, 2, 3) \)
\( B(x_2, y_2, z_2) = (4, 6, 3) \)
Calculate the difference for each coordinate:
- \( \Delta x = 4 - 1 = 3 \)
- \( \Delta y = 6 - 2 = 4 \)
- \( \Delta z = 3 - 3 = 0 \)
Now, plug these into the distance formula:
\[ d = \sqrt{(3)^2 + (4)^2 + (0)^2} \]
Calculate the squares:
\[ d = \sqrt{9 + 16 + 0} \]
Sum the values:
\[ d = \sqrt{25} \]
Take the square root:
\[ d = 5 \]
Step 4: Final Answer:
The distance between point \( A \) and point \( B \) is \( 5 \).
This matches Option (B).