How Do You Find a Vector Given Two Points?

To find a vector given two points, you subtract the coordinates of the initial point from the coordinates of the terminal point.

Here's a more detailed explanation:

Let's say you have two points in a 2D or 3D space:

• Initial Point (Starting Point): (A = (x_1, y_1, z_1))
• Terminal Point (Ending Point): (B = (x_2, y_2, z_2))

The vector from point A to point B, denoted as (\overrightarrow{AB}), is found by subtracting the coordinates of A from the coordinates of B:

(\overrightarrow{AB} = (x_2 - x_1, y_2 - y_1, z_2 - z_1))

In simpler terms:

1. Find the difference in the x-coordinates: (x_2 - x_1)
2. Find the difference in the y-coordinates: (y_2 - y_1)
3. Find the difference in the z-coordinates: (z_2 - z_1) (If it's a 3D vector)
4. Combine these differences to form the vector.

Example:

Let (A = (1, 2, 3)) and (B = (4, 5, 6)). Then the vector (\overrightarrow{AB}) is:

(\overrightarrow{AB} = (4 - 1, 5 - 2, 6 - 3) = (3, 3, 3))

So, the vector (\overrightarrow{AB}) is ((3, 3, 3)).

Key Points:

• The order of subtraction is crucial. You always subtract the initial point's coordinates from the terminal point's coordinates.
• The resulting vector represents the direction and magnitude of the displacement from the initial point to the terminal point.
• If you only have two points, this is the vector between those two points. If you require a unit vector, you need to divide the vector components by the magnitude of the vector.