SumNinja

Add and subtract vectors one part at a time

About 3 min

Add matching parts to join moves. To undo a move, reverse both of its parts and add that reverse.

A game piece goes two right and one down, then three left and four up. You want one move with the same end point. The vectors are (2, −1) and (−3, 4). Combine left-right changes with each other and up-down changes with each other.

Put the second move at the end of the first

A goes first, then B starts where A ends. C joins the same start and finish in one move.

Replace two moves with one

Combine (2, −1) and (−3, 4).

  1. Join the left-right changes

    2+(−3)=−12+(-3)=-1

    Two right followed by three left leaves one left. The first part of the answer is minus one.

  2. Join the up-down changes

    −1+4=3-1+4=3

    One down followed by four up leaves three up. That is the second part.

  3. Read the overall move

    [2−1]+[−34]=[−13]\begin{bmatrix}2\\-1\end{bmatrix}+\begin{bmatrix}-3\\4\end{bmatrix}=\begin{bmatrix}-1\\3\end{bmatrix}

    One left and three up reaches the same end as both moves together. The answer still has two parts.

Subtraction means undo the second move

Suppose the total move is (5, 2), and one part was (2, −1). Subtract that part to find the other move. Reverse it to (−2, 1), then add: (5, 2) + (−2, 1) = (3, 3).

Both signs reverse when you undo the move. In the up-down part, 2 − (−1) becomes 2 + 1. Taking away a downward move leaves one more unit upward.

Undo B by adding its reverse

A is the full move. The arrow labelled minus B reverses the part you remove. C is the remaining move, three right and three up.

Try it yourself

Work out an answer and explain your reasoning. Then check the explanation.

Combine two new moves

[−23]+[5−4]\begin{bmatrix}-2\\3\end{bmatrix}+\begin{bmatrix}5\\-4\end{bmatrix}

Which one move reaches the same end as these two?

Show the answerHide the answer
[3−1]\begin{bmatrix}3\\-1\end{bmatrix}

Left to right, −2 + 5 = 3. Up and down, 3 − 4 = −1.

Undo a signed move

[1−3]−[−24]\begin{bmatrix}1\\-3\end{bmatrix}-\begin{bmatrix}-2\\4\end{bmatrix}

Find the remaining move. Explain which direction the subtracted move must take when you undo it.

Show the answerHide the answer
[3−7]\begin{bmatrix}3\\-7\end{bmatrix}

Work with matching numbers: 1 − (−2) = 3 and −3 − 4 = −7.

Join or undo a fresh pair of moves

Choose addition or subtraction. Keep each sign with its number and work with matching parts. Your answer needs both changes.

2-minute practice · keep going after mistakes · opens in a new tab

Try vector addition Try vector subtraction