Vector Addition Calculator

What vector addition does

A vector carries a size and a direction, and adding two of them means placing the second arrow at the tip of the first and drawing a new arrow from the start of the first to the end of the second. OpenStax College Physics calls this the head to tail rule, and the parallelogram that falls out of it gives the same answer read the other way.

The arithmetic shortcut is that perpendicular components behave like separate problems. Motion east does not change motion north, so the x parts add among themselves and the y parts add among themselves. Two additions and one square root replace a drawing, a ruler and a protractor.

Order does not matter. Adding B to A gives the same resultant as adding A to B, which is why the parallelogram closes either way. Neither does the grouping: three vectors can be summed in any order, which is handy when two of them are easy to combine first.

What does not work is adding the magnitudes. A 3 unit vector east and a 4 unit vector north have a sum of 5 units, because the two are at right angles and the resultant is the hypotenuse. Only vectors pointing the same way have a sum of magnitudes equal to the sum of the parts. NASA's Glenn Research Center works the same rule through a worked aircraft example.

The component method in four steps

Every vector splits into an x part and a y part using the angle it makes with the positive x axis. The x component is the magnitude times the cosine of the angle, and the y component is the magnitude times the sine.

Step 1 is where quadrant matters. The cosine of 120 degrees is -0.5000, so an 8 unit vector at 120 degrees moves 4 units in the negative x direction rather than positive. A calculator in degree mode returns that sign automatically; a hand calculation that drops it puts the resultant in the wrong quadrant and keeps the right magnitude.

Step 4 reverses the process, and the two operations are not symmetric in their accuracy. Magnitudes add in quadrature, so a small relative error in one component barely moves the result. Angles are more fragile, because a resultant with a small x component swings through a wide range of angles for a small change in y.

A worked sum: 10 at 30 degrees plus 8 at 120 degrees

Take A as 10 units at 30 degrees and B as 8 units at 120 degrees. The table shows the components and the totals.

The components add to 4.660 across and 11.928 up. Squaring them gives 21.716 and 142.277, and 21.716 + 142.277 = 163.993. The square root of 163.993 is 12.806, which is the magnitude. The direction is the inverse tangent of 11.928 divided by 4.660, and that ratio is 2.5597, whose inverse tangent is 68.66 degrees. Both components are positive, so the angle needs no adjustment.

A drawing confirms the size. The two vectors sit close to a right angle: the angle between them is 90 degrees exactly, since 120 minus 30 is 90. For perpendicular vectors the magnitude of the sum is the square root of the sum of the squares of the magnitudes, and the square root of 100 + 64 = 164 is 12.806. Two routes, one answer.

Quadrants and the inverse tangent trap

Inverse tangent returns an angle between -90 and 90 degrees, which covers only half of the circle. The other half has to be recovered from the signs of the components, or from a two argument arctangent function if the software offers one.

The example above has both components positive, so 68.66 degrees stands. Flip the sign of the x component to -4.660 and keep 11.928 up: the ratio becomes -2.5597 and inverse tangent returns -68.66 degrees, which points into the fourth quadrant. Adding 180 degrees gives 111.34 degrees, the correct second quadrant answer.

Software that has atan2 avoids the table. Give it y first and x second, and it reads the signs directly: atan2(11.928, -4.660) returns 111.34 degrees in one call. Passing x first is the classic bug, and it shows up as an answer reflected across the 45 degree line.

Subtraction and the other diagonal

Subtracting a vector means adding its negative, which flips both components. Subtracting B from A adds 4.000 to the x component, taking it from 8.660 to 12.660, and takes the y component from 5.000 down to -1.928.

That difference has magnitude 12.806 as well, because 12.660 x 12.660 = 160.276 and 1.928 x 1.928 = 3.717, and 160.276 + 3.717 = 163.993. Its direction is the inverse tangent of -1.928 divided by 12.660, which is -8.66 degrees, or 351.34 degrees as a compass bearing from the x axis.

The equal magnitudes are not a coincidence. A and B are perpendicular here, since their dot product comes to zero: 8.660 times -4.000 gives -34.64, 5.000 times 6.928 gives 34.64, and the two terms cancel. For perpendicular vectors the parallelogram is a rectangle, and the two diagonals of a rectangle are the same length. Tilt either vector and the two diagonals separate: the sum becomes the longer one and the difference the shorter.

Relative displacement uses the same operation. If a drone moves 6 km east then 2 km north, and a second drone moves 3 km east then 5 km north, the displacement from the second to the first is 3 km east and 3 km south, a distance of 4.243 km on a bearing of 135 degrees.

Three vectors that close a loop

Three vectors of equal length at 0, 120 and 240 degrees add to zero. Each pair is 120 degrees apart, so the three spokes of a wheel pull evenly in all directions and cancel.

A zero resultant has no direction, and that is the correct answer rather than a missing one. A calculator that divides by a zero x component while computing the angle will return infinity or an error, and the honest output is that the vectors are in balance.

This case shows up in rigging and in electrical work. Three ropes at 120 degrees hold a load with no net sideways pull, and three phase power sums to zero in the same geometry. Add a fourth vector and the balance usually breaks: a 10 unit vector at 0 degrees combined with the three above leaves the 10 units at 0 degrees unopposed, because the other three already cancel.

Speed, velocity and the river crossing

Speed is a magnitude and velocity is a vector, so the two combine differently. A boat pointed north at 4 m/s on a river flowing east at 3 m/s travels at 5 m/s over the ground, at 36.87 degrees east of north.

The components make it plain: 4 m/s north is (0, 4) and 3 m/s east is (3, 0), so the sum is (3, 4), whose magnitude is 5 and whose angle is the inverse tangent of 3 over 4, which is 36.87 degrees measured from the north axis. Steering 36.87 degrees upstream of the intended course cancels the current and puts the boat back on a due north track.

Time to cross depends only on the northward component. A 60 m wide river crossed at 4 m/s north takes 15 seconds no matter how fast the current runs, because the current moves the boat downstream rather than across. That independence of perpendicular components is the same rule as the component method, read as a motion problem.

Three dimensions extend the method with one more component. A vector from the origin to the point (2, 3, 6) has magnitude 7, since 4 + 9 + 36 = 49 and the square root of 49 is 7. Direction then needs two angles rather than one, and most software reports the components instead because they are unambiguous.

Mistakes that survive a sanity check

Four errors produce answers that look reasonable and are wrong.

Adding magnitudes is the first. Two 10 unit vectors at 90 degrees add to 14.142, not 20, and two 10 unit vectors pointing opposite ways add to 0. The magnitude of a sum is bounded by the sum of the magnitudes and by their difference, so any result outside that range is a sign error.

Dropping the sign of a component is the second. The cosine of an angle between 90 and 270 degrees is negative, and a layout that treats every x component as positive puts the resultant in the first quadrant when it belongs in the third. The magnitude stays right and the direction is off by up to 180 degrees, which is exactly the kind of error that passes a quick look.

Mixing degrees and radians is the third. A spreadsheet given 120 in a radians cell returns a cosine of 0.8142 rather than -0.5000, and every downstream number follows it.

Swapping the component order is the fourth, and it is a units problem more than a maths one. Physics courses often write (north, east) while maths courses write (x, y), so a vector copied between the two conventions comes out reflected across the diagonal. Write the axes down before typing the numbers.