How this instrument works
A linear combination takes two vectors, stretches or shrinks each one by its own scalar, and lays the results end to end: x = c₁v₁ₓ + c₂v₂ₓ and y = c₁v₁ᵧ + c₂v₂ᵧ. Ordinary vector addition is just the special case c₁ = c₂ = 1; this sheet generalises that to any pair of weights, which is exactly the operation underneath every basis, span, and coordinate system in linear algebra.
The two-step order is not arbitrary. Scaling first fixes how far along each vector's own direction you travel — double c₁ and you double that vector's contribution without touching the other's; only once both stretches are set does addition combine them, walking c₁ copies of v₁ then c₂ copies of v₂ tip to tail. Numerically the order of addition never matters, but thinking of it as 'scale, then place' is what makes the geometry click.
Two edge cases show why the idea matters beyond a single answer. Set a scalar to zero and its vector vanishes from the sum entirely, leaving only the other one untouched. Set the second vector to the exact negative of the first with matching scalars and the combination collapses to (0, 0) no matter how large the vectors were — a vector always cancels its own opposite. When v₁ and v₂ point in genuinely different directions, letting c₁ and c₂ range over every real number reaches every point in the plane; the pair is then said to span it.
- Enter the first vector's components into 'v₁: x' and 'v₁: y', then set its weight in 'Scalar c₁'.
- Do the same for the second vector: fill 'v₂: x' and 'v₂: y', weighted by 'Scalar c₂'.
- Read the combined vector directly off 'Result: x' and 'Result: y' — no further arithmetic needed.
- To see cancellation or a dropped term, set a scalar to 0 or make one vector the exact negative of the other and watch the result respond instantly.
Worked example — combining (1, 2) and (4, 5)
Scale v₁ = (1, 2) by c₁ = 2 to get (2, 4); scale v₂ = (4, 5) by c₂ = 3 to get (12, 15). Adding those two scaled vectors component by component gives x = 2 + 12 = 14 and y = 4 + 15 = 19, so 2(1,2) + 3(4,5) lands at (14, 19) — the same figures this sheet returns for 'Result: x' and 'Result: y' with those six inputs.
Picture v₁ and v₂ as two thruster directions on a small drone, each firing at a different duty cycle: c₁ = 2 means the first thruster fires twice as hard, c₂ = 3 means the second fires three times as hard. The drone's net displacement is the linear combination of the two, (14, 19) — the point (2,4) reached by the first thruster alone, shifted further by the (12,15) contributed by the second.
Questions
What is a linear combination of two vectors?
It is the vector formed by scaling each vector by its own number (a scalar, or weight) and then adding the results component by component: c₁v₁ + c₂v₂. With c₁ = 2, v₁ = (1,2), c₂ = 3, v₂ = (4,5), that reads 2(1,2) + 3(4,5) = (2,4) + (12,15) = (14,19).
How is a linear combination different from ordinary vector addition?
Ordinary addition is the special case where both scalars equal 1: v₁ + v₂. A linear combination lets each vector carry its own weight before the vectors are added, so c₁ = 2 and c₂ = 3 stretch v₁ and v₂ by different amounts first — plain addition is just one point on a much larger family of combinations.
What happens if one scalar is zero?
That vector's entire contribution disappears — c₁ = 0 leaves the result equal to c₂v₂ exactly, since anything multiplied by zero vanishes. Using c₁ = 0 with v₂ = (4,5) returns (4,5) untouched, as if v₁ were never entered.
Why does the combination equal the zero vector when v₂ is the negative of v₁?
Scaling v₁ by 1 and its own negative by −1 produces two vectors of equal length pointing opposite ways, and adding opposite vectors always cancels to (0,0), whatever their length. Try c₁ = 1, v₁ = (3,3), c₂ = −1, v₂ = (3,3) — the result lands exactly on the origin.
How does a linear combination relate to the span of a set of vectors?
The span of v₁ and v₂ is the full set of points reachable by every possible choice of c₁ and c₂. When the two vectors point in different directions, that set is the entire plane; if they happen to be parallel, every combination stays confined to a single line no matter which scalars are chosen.
Can the scalars be negative or fractional rather than whole numbers?
Yes — c₁ and c₂ may be any real number: positive, negative, zero, or a fraction. A negative scalar reverses that vector's direction before adding; a fractional scalar shrinks it. The formula x = c₁v₁ₓ + c₂v₂ₓ, y = c₁v₁ᵧ + c₂v₂ᵧ handles every case identically, with no special rule for non-integers.