Vector Addition Calculator

A vector carries a size and a direction, so it is stored as components — how far across, how far up:

$\mathbf{a} + \mathbf{b} = (a_x + b_x,\; a_y + b_y)$

Add the $x$ parts together and the $y$ parts together, and never mix the two. In 3D there is simply a third slot: $(a_x + b_x,\; a_y + b_y,\; a_z + b_z)$.

The result is called the resultant. Its size comes from Pythagoras:

$|\mathbf{r}| = \sqrt{r_x^2 + r_y^2}$

and its direction from $\theta = \tan^{-1}(r_y / r_x)$, measured anticlockwise from the positive $x$-axis. Because $\tan^{-1}$ only returns angles between $-90°$ and $90°$, add $180°$ whenever $r_x$ is negative.

Components are what make vectors easy: two directions that would be awkward to combine geometrically become two ordinary sums of numbers. Any number of vectors can be added at once by totalling each column, and the order you add them in never matters.

A vector given as "$5$ at $30°$" cannot be added directly. Break each one into components, add, then convert back:

$a_x = |\mathbf{a}| \cos\theta, \qquad a_y = |\mathbf{a}| \sin\theta$

So $5$ at $30°$ becomes $(5\cos 30°,\; 5\sin 30°) = (4.330,\; 2.500)$.

Do this for every vector, sum the columns, and rebuild the magnitude and angle from the totals. Three things to watch:

• Degree mode, unless the angle is genuinely in radians.
• Signs come from the quadrant. At $120°$, cosine is negative, so $8\cos 120° = -4$.
• Keep the unrounded components through the addition and round only the final magnitude and angle.

Drawn on paper, adding vectors is tip to tail: place the start of $\mathbf{b}$ at the end of $\mathbf{a}$, and the resultant runs from the very start to the very end. Order does not matter, $\mathbf{a} + \mathbf{b} = \mathbf{b} + \mathbf{a}$, which is why the parallelogram picture works too.

Subtraction is adding the reverse. $\mathbf{a} - \mathbf{b} = \mathbf{a} + (-\mathbf{b})$: negate every component of $\mathbf{b}$, then add as usual.

The big mistake is adding the magnitudes. A vector of size $5$ plus a vector of size $8$ almost never gives $13$ — that only happens if they point the same way. At right angles the result is $\sqrt{89} \approx 9.43$, and pointing opposite ways it is $3$. The resultant of two vectors always lies between $|\,|\mathbf{a}| - |\mathbf{b}|\,|$ and $|\mathbf{a}| + |\mathbf{b}|$.

A quick check on any resultant: it must be no longer than the two lengths added together, and no shorter than their difference. If your answer falls outside that window, a sign or an angle has gone astray somewhere in the components.