5 mistakes learners make with vector arithmetic, each one named and explained.
Scalars multiply every component, and the components stay separate throughout.
2p doubles the first component and leaves the second as it was.
The multiplier is written once, to the left, so it looks as though it acts once.
Question. p = (5, -3), find 2p
A common answer. (10, -3)
The answer. (10, -6)
Why. The multiplier acts on the whole vector, so both parts double.
A scalar multiplies everything inside the bracket, exactly as in algebra.
a + 2b is worked out as a + b, with the 2 never used.
The letters are what the expression is about and the small number in front is easy to read past.
Question. a = 3i - 2j, b = i + 4j, find a + 2b
A common answer. 4i + 2j
The answer. 5i + 6j
Why. 2b is 2i + 8j, so both components of b double before adding.
Work out 2b as its own vector on a separate line, then add.
Position after 3 seconds is found by multiplying both the start and the velocity by 3.
The 3 is the number the question turns on, so it gets applied to everything in sight.
Question. Start at 4i - 3j, velocity 2i + 5j, after 3 seconds
A common answer. 3(4i - 3j) + 3(2i + 5j)
The answer. (4i - 3j) + 3(2i + 5j)
Why. The starting position happens once; only the velocity accumulates with time.
Only the rate gets multiplied by the time. The starting point is added exactly once.
a minus b is worked out as b minus a, so every sign is reversed.
Subtraction order is invisible in a column of numbers and both orders look equally plausible.
Question. a = (2, 5), b = (7, 1), find a - b
A common answer. (5, -4)
The answer. (-5, 4)
Why. The first vector named is the one you start from.
The two answers are exact opposites. If yours has every sign flipped, the order is reversed.
The components are subtracted when the question asked for a sum.
Vector questions come in matched pairs of adding and subtracting and the sign is one small symbol.
Question. (3, -2) + (1, 4)
A common answer. (2, -6)
The answer. (4, 2)
Why. Adding means both components go up by the second vector.
Read the sign in the question aloud before starting. It is the only thing that differs.
WAJD spots these patterns in your child's answers and names the one behind their wrong answers, instead of just marking them wrong.