3 mistakes learners make with parallel vectors, each one named and explained.
Parallel means one is a scalar multiple of the other, in every component at once.
(2, -3) is chosen as parallel to (2, 3).
Minus one really is a scalar, so changing signs feels like it should qualify.
Question. Parallel to (2, 3)
A common answer. (2, -3)
The answer. (4, 6)
Why. Only the second component changed sign, so the two components were not scaled by the same number.
Multiplying by a negative is allowed, but it has to be the same negative for both components.
(6, 8) is chosen as parallel to (2, 3) because the first component is three times as big.
Checking one component is quick and it is right often enough to feel reliable.
Question. Parallel to (2, 3)
A common answer. (6, 8)
The answer. (4, 6)
Why. 2 to 6 is a multiplier of 3, but 3 to 8 is not.
Both components have to be multiplied by the same number. Check both, every time.
k is found by adding the difference between the known components.
Adding a constant is the other way two numbers can be related, and it is the more familiar one.
Question. 2i + kj parallel to 6i - 9j
A common answer. k = -6, from a difference of 4
The answer. k = -3, from a multiplier of 3
Why. Parallel means a scalar multiple, and a scalar multiplies rather than adds.
Parallel is always multiplicative. A difference would change the direction.
WAJD spots these patterns in your child's answers and names the one behind their wrong answers, instead of just marking them wrong.