Does moving an arrow change the vector?

Suppose a robot gets the instruction move three squares right and two squares up. Does the instruction change when the robot starts somewhere else?

You need coordinate pairs and subtraction. In $(x,y)$, the first number gives horizontal position and the second gives vertical position. Positive directions are right and up; negative directions are left and down.

Keep a record of your investigation

Use $(0,0)\to(3,2)$ as your reference. In a notebook or text file, write the predicted endpoint before each move. Keep the movement $(3,2)$ fixed; change only the starting point. Record the start, finish, and finish minus start for each trial. Compare each difference with the reference before explaining what a vector records.

Move the start, predict the finish

Starting at $(0,0)$, the robot finishes at $(3,2)$. Before using the explorer, predict its finish if it starts at $(-2,1)$. Which numbers should change: the start, the finish, or the movement?

Select Start at (−2, 1). Then predict the result of Start at (1, −3) and try it. The dashed arrow shows the original movement from the origin for comparison.

Describe what both arrows have in common. Use the horizontal and vertical changes, rather than their endpoint coordinates.

Record the change, not just the finish

A displacement vector records a change in position. If the starting point is $P=(p_x,p_y)$ and the endpoint is $Q=(q_x,q_y)$, then

For the first moved arrow,

This is why an arrow ending at $(1,3)$ can represent the vector $(3,2)$. Endpoint coordinates describe a position; subtracting the start gives the movement.

When the start is the origin, the endpoint and displacement happen to have the same coordinates. That convenient drawing is called standard position. It does not mean every arrow must start there.

Why translation preserves the vector

Imagine adding a fixed shift $\mathbf s$ to both endpoints. Their difference becomes

The added shift cancels. This argument explains every translation, including ones outside the explorer, rather than only the two cases we tried.

We write vectors horizontally in this lesson for compactness. Later, the same vector can be written as a column:

These are two notations for the same two components. Neither introduces a new movement.

A different start on paper

An arrow goes from $P=(4,-2)$ to $Q=(1,3)$. Find its displacement. Now shift both points by $(2,-1)$. Predict the new displacement before calculating the new endpoints.

For a data record, a vector can similarly describe a change: new measurements minus old measurements. Keep track of whether a tuple represents the record itself or an update to that record.

Design your own comparison

Choose your own starting point within the explorer’s range. Keep its vertical coordinate fixed and change only its horizontal coordinate. Write a prediction, test it, and record both endpoints and their difference.

Now design a paper comparison that does change the displacement: move only one endpoint. Choose the amount yourself. Explain why this experiment differs from translating the whole arrow.

Write a short conclusion: what you predicted, what you observed, and how the evidence changed or supported your explanation.

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