Vector Addition and Subtraction Calculator – Geometric & Coordinate Methods

The calculator above adds and subtracts two vectors in two ways. The Geometric Method tab is for problems that give the magnitudes of both vectors and the angle between them: it applies the parallelogram rule and the law of cosines to find |a + b| and |a − b|, offers quick buttons for 0°, 30°, 45°, 60°, 90°, 120°, 135°, 150° and 180°, and shows exact surd answers for those special angles. The Coordinate Method tab is for problems given in coordinates in the Oxy plane or Oxyz space: it adds and subtracts component by component and also finds the magnitudes, the dot product and the angle between the vectors. Every result includes a to-scale diagram and a step-by-step solution you can compare with your work, copy, or print on A4 paper.

📐 Quick refresher: what is a vector?

A vector is a directed line segment. The vector from point A to point B is written AB; when the endpoints do not matter we simply write a, b. The magnitude (length) of a vector is the distance between its tail and its tip: |AB| = AB. Two vectors are equal when they have the same direction and the same magnitude, so a vector can be slid to start at any point without changing it. A vector whose tail and tip coincide is the zero vector 0, with magnitude 0.

The angle between two non-zero vectors is measured with both vectors drawn from a common tail: draw OA = a, OB = b; then (a, b) = angle AOB, which always lies between 0° and 180°.

➕ Adding two vectors

1. Definition and the triangle rule

Given vectors a and b, pick any point A, draw AB = a, then draw BC = b starting at the tip of the first one. The vector AC is the sum of the two vectors. This gives the triangle rule (three-point rule), true for any three points A, B, C:

Picture walking from A to B and then from B to C: the overall result is the same as walking straight from A to C. The rule extends to a chain of points: AB + BC + CD = AD.

2. The parallelogram rule

When the two vectors share a common tail, use the parallelogram rule: if ABCD is a parallelogram, then

The sum of two vectors with a common tail is the diagonal of the parallelogram built on them, starting at the common tail. In physics this is exactly how the resultant of two concurrent forces is found. The Geometric tab of the calculator draws this parallelogram OACB for every problem.

3. Properties of vector addition

➖ Subtracting two vectors

The opposite of a, written −a, has the same magnitude as a but points the other way. For example, the opposite of AB is BA. The difference of two vectors is defined as adding the opposite:

This gives the subtraction rule: for any three points O, A, B,

How to remember it: the difference of two vectors with a common tail joins their tips and points toward the tip of the first vector. In parallelogram OACB above, the sum is the diagonal OC and the difference is the diagonal BA, so the magnitudes of the sum and the difference are simply the lengths of the two diagonals.

🔢 Geometric method: magnitude of the sum and difference

When the magnitudes |a|, |b| and the angle α = (a, b) are known, square the sum (or difference) as a dot product and use a·b = |a||b|cos α:

This is the law of cosines applied to triangles OAB and OAC inside the parallelogram. For the difference, the angle between sides OA and OB is exactly α, which gives the minus sign. For the sum, the angle at vertex A of triangle OAC is 180° − α, and since cos(180° − α) = −cos α, the minus turns into a plus.

Special cases

When |a| = |b|, the parallelogram is a rhombus, so its diagonals are perpendicular: (a + b) ⊥ (a − b). If in addition α = 60°, triangle OAB is equilateral, so |a − b| = |a| and |a + b| = |a|√3.

Two identities give a quick check: the parallelogram law |a + b|² + |a − b|² = 2(|a|² + |b|²) and the triangle inequality ||a| − |b|| ≤ |a ± b| ≤ |a| + |b|. If your answer breaks either one, something went wrong.

📊 Coordinate method: add and subtract component by component

1. In the Oxy plane

Let a = (x₁; y₁) and b = (x₂; y₂). To add or subtract the vectors, add or subtract the matching coordinates:

The companion formulas you usually need in the same problem:

2. In Oxyz space

For a = (x₁; y₁; z₁) and b = (x₂; y₂; z₂), the formulas keep the same shape with a third component:

The coordinates of AB are the tip coordinates minus the tail coordinates: AB = (x_B − x_A; y_B − y_A). This is itself a vector subtraction, because AB = OB − OA.

💡 Worked examples

Example 1. Perpendicular vectors

Two perpendicular vectors have |a| = 3, |b| = 4. Find the magnitudes of their sum and difference.

The parallelogram is a 3 × 4 rectangle, so both diagonals are equal to 5 (the 3-4-5 Pythagorean triple).

Example 2. A 60° angle

Let |a| = 3, |b| = 4 with an angle of 60° between them.

Check with the parallelogram law: 37 + 13 = 50 = 2(9 + 16). This is also the built-in example of the Geometric tab.

Example 3. Sum and difference in the Oxy plane

Let a = (2; −3) and b = (−5; 1).

Watch the signs: subtracting a negative number means adding its opposite, 2 − (−5) = 7.

Example 4. The angle between two vectors

Let a = (1; 2) and b = (3; −1). Find the angle between them.

Example 5. In Oxyz space

Let a = (1; 2; −1) and b = (2; −1; 3).

The dot product is negative, so the angle between the vectors is obtuse: α ≈ 109.11°.

Example 6. Resultant of two forces

Forces of 30 N and 40 N act on an object at an angle of 60° to each other. The magnitude of the resultant is:

🧭 Where vector addition is used

• Combining forces: the resultant of concurrent forces is their vector sum; the pull of two cables or two people towing one object is found with the parallelogram rule.
• Relative velocity: a boat moving at 4 m/s straight across a river that flows at 3 m/s actually travels at √(4² + 3²) = 5 m/s relative to the bank.
• Displacement: walking 3 km east and then 4 km north leaves you 5 km from the start as the crow flies, which is the magnitude of the sum vector.
• Computer graphics and games: a character’s new position is the old position plus velocity times time; the direction from the character to a target is the difference of two position vectors.
• Maps and navigation: the vector joining two points on a map is the difference of their position vectors, and its magnitude is the distance between them.

🎓 Tips for solving vector problems quickly

1. Pick the right rule. Vectors placed tip to tail (A → B → C): use the triangle rule. Vectors with a common tail: use the parallelogram rule. Coordinates given: use the coordinate method.
2. Insert a point. Any point can be inserted in the middle: AB = AM + MB = OB − OA.
3. Know the classic identities: I is the midpoint of AB if and only if IA + IB = 0, and then MA + MB = 2MI for every point M. G is the centroid of triangle ABC if and only if GA + GB + GC = 0.
4. Check with the triangle inequality: the magnitude of a sum or difference always lies between ||a| − |b|| and |a| + |b|.

⚠️ Common mistakes

1. Adding magnitudes instead of vectors. |a + b| = |a| + |b| only holds when the vectors point the same way; for perpendicular vectors of lengths 3 and 4 the sum has length 5, not 7.
2. Reversing the difference. OA − OB = BA, not AB.
3. Using the wrong angle. The angle between vectors is measured with a common tail. For vectors placed tip to tail, AB, BC, the angle between them is 180° − angle ABC.
4. Dropping the sign of cos for obtuse angles. cos 120° = −1/2, so for an obtuse angle the sum is shorter than the difference.
5. Sign errors with negative coordinates. Always wrap negative numbers in brackets: −3 − (−2) = −1.
6. Rounding too early. Keep surds or plenty of decimals until the final step to avoid accumulated error.

❓ Frequently asked questions (FAQ)

How do the geometric and coordinate methods differ?

The geometric method is used when you know the magnitudes of the two vectors and the angle between them: apply |a ± b|² = |a|² + |b|² ± 2|a||b|cos α to get the magnitude of the sum or difference. The coordinate method is used when coordinates are given: add or subtract component by component to get both the coordinates and the magnitude of the result. The coordinate method is easier to compute, the geometric method is more visual.

When are two vectors perpendicular?

Two non-zero vectors are perpendicular if and only if their dot product is 0. In Oxy: x₁x₂ + y₁y₂ = 0; in Oxyz: x₁x₂ + y₁y₂ + z₁z₂ = 0. In that case |a + b| = |a − b| = √(|a|² + |b|²).

What is the difference between a − b and b − a?

They are opposite vectors: b − a = −(a − b). They have the same magnitude but point in opposite directions, and every coordinate changes sign. For example, if a − b = (7; −4) then b − a = (−7; 4).

What does a negative dot product mean?

The sign of the dot product tells you the type of angle between the vectors: a·b > 0 means acute, a·b = 0 means right, a·b < 0 means obtuse (between 90° and 180°). For an obtuse angle the difference is always longer than the sum.

Can the sum of two vectors be shorter than each of them?

Yes. When the angle between the vectors is large enough, the sum can be shorter than both. For example, with |a| = |b| = 1 and an angle of 150°, |a + b| = √(2 − √3) ≈ 0.52. The magnitude of the sum always lies between ||a| − |b|| and |a| + |b|.

Does this calculator compute the cross product?

This calculator focuses on the sum, difference, magnitudes, dot product and angle of two vectors. For the cross product a × b in Oxyz space, use the 3D vector calculator on ATMath, linked in the Related tools section below.

🔗 Related tools

• 3D vector calculator – dot & cross product
• Triangle solver – law of sines & cosines
• Right triangle side calculator
• Triangle area calculator