Column vectors and reversing: reversing (−3, 7) gives (3, −7)

Here is a canonical AQA Foundation trigger (reverse a column vector):

The misconception is to swap the numbers and write $(7,\ -3)$. But going from B back to A is the same journey reversed, so each step turns around: the horizontal and vertical components keep their positions and only change sign.

Negate each component in place: $-3 \to 3$ and $7 \to -7$, giving the column vector $(3,\ -7)$.

A column vector looks like two numbers stacked in brackets, and that layout invites two slips. The first is to “reverse” by turning the stack upside down, swapping top and bottom to get $(7,\ -3)$. But the top number is the horizontal step and the bottom is the vertical step; they are not interchangeable, and swapping them describes a completely different journey.

Thinking about direction fixes it. Reversing a vector means walking the same path backwards, so every step turns around: 3 left becomes 3 right, 7 up becomes 7 down. Each component keeps its row and only changes sign. So $(-3,\ 7)$ reversed is $(3,\ -7)$. The diagram below shows the original arrow from A to B and its reverse from B to A sitting on top of it, pointing the opposite way.

The second slip is to read the bracket as a fraction. Adding $(1,\ 2)$ and $(4,\ 6)$ tempts a student to combine across the line and write $\tfrac76$. But the components never mix: add tops to tops and bottoms to bottoms, so $1 + 4 = 5$ over $2 + 6 = 8$, giving the column vector $(5,\ 8)$.

Flip the signs, keep the positions. Reversing a vector negates the top and the bottom where they sit. For $(-3,\ 7)$: $-3 \to 3$ on top and $7 \to -7$ below, giving $(3,\ -7)$.

Don't swap the rows. The top is the horizontal step and the bottom is the vertical step. Swapping them gives $(7,\ -3)$, a different vector that points a different way.

Add and subtract row by row. A column vector is not a fraction. To add $(1,\ 2)$ and $(4,\ 6)$, add tops then bottoms: $1 + 4 = 5$ and $2 + 6 = 8$, so the sum is $(5,\ 8)$ — not $\tfrac76$.

Write it as a column vector. Give the answer as two stacked components, not as a point's coordinates or a fraction. Reversing $(-3,\ 7)$ is the column vector $(3,\ -7)$.

Reverse the vector $(-3,\ 7)$, then add $(1,\ 2)$ and $(4,\ 6)$. The trap is to swap rows or merge them into a fraction; the fix is to work component by component.

1. Negate each component to reverse. Flip the sign of the top and of the bottom, keeping their rows.
   $$(-3,\ 7) \;\longrightarrow\; (3,\ -7)$$
2. Reject the swap. Swapping rows gives $(7,\ -3)$, which moves 7 right and 3 down — a different journey from the 3 right and 7 down of the true reverse.
3. Add row by row. Tops to tops, bottoms to bottoms.
   $$(1,\ 2) + (4,\ 6) = (1 + 4,\ \ 2 + 6) = (5,\ 8)$$
4. Don't make a fraction. The bracket is two stacked components, so the answer is the column vector $(5,\ 8)$ — never $\tfrac76$.

So $(-3,\ 7)$ reversed is $(3,\ -7)$ (negate in place, not $(7,\ -3)$), and $(1,\ 2) + (4,\ 6) = (5,\ 8)$, each component handled on its own row.