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Vector Subtraction (Component Form)

cx=ax−bxc_x = a_x - b_x

Worked example: 7 minus (−3) → 10 — press Try an example to run it live, then adjust anything.

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Vector Subtraction (Component Form) explained

bxaxcx

Vectors subtract exactly as they add — one axis at a time, with no interaction between components. Apply this formula once per axis and you have the whole difference vector. The result a − b is the arrow that runs from the tip of b to the tip of a, which is why it answers every "relative to" question: an aircraft flying at vplane through wind vair has a ground velocity found by subtraction, and a police radar reports the difference between two velocity vectors rather than either one alone.

Order is everything. a − b and b − a have the same length but point opposite ways, and swapping them is the standard way to get a relative-velocity answer that is right in size and backwards in direction. A quick check: if ax = 7 and bx = −3, then cx = 7 − (−3) = 10 — subtracting a negative component moves the difference further out, not closer in.

Vector Subtraction (Component Form) formula

cx=ax−bxc_x = a_x - b_x
Where
  • cxc_x= Component of the difference a − b
  • axa_x= Component of a
  • bxb_x= Component of b

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