SumNinja

Scale or reverse a vector

About 3 min

Multiply both parts by the same number to resize a move. A minus sign also turns it around.

A game move goes two right and three down: (2, −3). You need to go twice as far along that direction, or turn back along the same line. Scaling changes both parts together. The one number you multiply by has a name: a scalar.

The reverse follows the same line

A is the original move. C is minus two times A: twice as long, in the opposite direction.

Stretch the move, then reverse it

Start with the move (2, −3).

  1. Repeat the move twice

    2[2−3]=[4−6]2\begin{bmatrix}2\\-3\end{bmatrix}=\begin{bmatrix}4\\-6\end{bmatrix}

    Double both parts. Four right and six down goes twice as far in the same direction.

  2. Reverse as well as stretch

    −2[2−3]=[−46]-2\begin{bmatrix}2\\-3\end{bmatrix}=\begin{bmatrix}-4\\6\end{bmatrix}

    The size two doubles the length. The minus sign reverses the direction, giving four left and six up.

  3. See what zero does

    0[2−3]=[00]0\begin{bmatrix}2\\-3\end{bmatrix}=\begin{bmatrix}0\\0\end{bmatrix}

    Both changes become zero, so the piece stays put. The zero vector has no direction.

A fraction can shrink the whole move

Half of (8, −2) is (4, −1). Both changes halve, so the piece stays on the same line and goes half as far. A positive fraction below one shrinks a vector.

Changing only one part can change the direction too. For (2, −3), doubling only the first part gives (4, −3). That is a new direction, not a doubled copy of the move.

The scalar is one number, but the answer remains a vector with two parts. This differs from a dot product, which combines two vectors into one number.

Try it yourself

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

Use a fractional scalar

12[6−4]\frac12\begin{bmatrix}6\\-4\end{bmatrix}

What vector gives half of this move?

Show the answerHide the answer
[3−2]\begin{bmatrix}3\\-2\end{bmatrix}

Halving both parts keeps the same direction and makes the move half as long.

Apply the factor everywhere

3[25]=[65]  ?3\begin{bmatrix}2\\5\end{bmatrix}=\begin{bmatrix}6\\5\end{bmatrix}\;?

The claimed result changes only one part. What is the correct full move?

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[615]\begin{bmatrix}6\\15\end{bmatrix}

Multiply the second part by three as well, so 5 becomes 15.

Choose the scale yourself

[2−3]⟶[−69]\begin{bmatrix}2\\-3\end{bmatrix}\quad\longrightarrow\quad\begin{bmatrix}-6\\9\end{bmatrix}

Which single scalar makes the change to both parts?

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−3-3

Both parts multiply by minus three. The result goes three times as far in the opposite direction.

Resize and reverse new moves

The game uses whole numbers to scale each move, with minus signs too. It does not test the fraction examples yet.

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

Try vector scaling