VecLab

Vector Calculator (2D)

The basic 2D vector operations, together.

x₁

y₁

x₂

y₂

Sum (x, y): (4, 6)
Dot product: 11
|v₁|: 5
|v₂|: 2.236
Angle between (°): 10.305

What you entered

v₁: (3, 4)v₂: (1, 2)

1
Vector sum (add component-wise)
(3+1, 4+2)= (4, 6)
2
Dot product
3×1 + 4×2= 11
3
Magnitude |v₁|
√(3² + 4²)= 5
4
Magnitude |v₂|
√(1² + 2²)= 2.2361
5
Angle between them
cos⁻¹(11 ÷ (5×2.2361))= 10.3048°

Result

Dot product: 11

v₁ and v₂ are 10.3° apart, with a dot product of 11 — a positive dot product means they broadly point the same way, negative means broadly opposite, and zero means perpendicular.

Sum, dot product and angle for common 2D vector pairs

Enter the four components x1, y1, x2 and y2 from a row and the calculator returns the same sum, dot product and separation angle. The first row is its default state.

v1 (x1, y1)	v2 (x2, y2)	Sum	Dot product	Angle between (deg)
(3, 4)	(1, 2)	(4, 6)	11	10.305
(1, 0)	(0, 1)	(1, 1)	0	90
(3, 0)	(0, 4)	(3, 4)	0	90
(3, 4)	(6, 8)	(9, 12)	50	0
(1, 1)	(-1, -1)	(0, 0)	-2	180
(10, 5)	(-3, 8)	(7, 13)	10	83.991
(2, 3)	(4, -1)	(6, 2)	5	70.346
(5, 12)	(3, 4)	(8, 16)	63	14.25

The two rows with a dot product of 0 are perpendicular pairs, which is the quickest test for a right angle between vectors. Row 4 is two vectors pointing the same way, so the angle is 0 even though one is twice as long, and row 5 is a pair pointing exactly opposite at 180 degrees. Magnitudes are also shown on the page: (3, 4) has length 5 and (5, 12) has length 13.

Quantities with direction, not just size

A vector encodes both magnitude and direction — unlike a plain number, (3, 4) isn't just 'size 5' (its magnitude via Pythagoras), it also points a specific way, which is why vector addition combines both components separately.

What the dot product tells you

The dot product measures how much two vectors point in the same direction — it's zero when they're perpendicular, positive when they broadly align, and negative when they point opposite ways, which is why it's central to physics work/force calculations.

Frequently asked questions

A drone flies 3 km east then 4 km north — how far is it from the start and at what angle?

The displacement vector is (3, 4). Magnitude = √(3² + 4²) = √25 = 5 km. Angle from east = arctan(4/3) ≈ 53.1°. The drone is 5 km from the start at about 53° north of east.

Two forces act on a bolt: (10, 5) N and (−3, 8) N. What's the net force?

Vector sum: (10 + (−3), 5 + 8) = (7, 13) N. Magnitude = √(49 + 169) = √218 ≈ 14.76 N. The calculator shows the component-wise addition and resultant magnitude.

How is the dot product different from the cross product?

The dot product gives a scalar (a single number) measuring alignment. The cross product gives a vector perpendicular to both inputs, measuring the area of the parallelogram they span. Dot product works in any dimension; the cross product is specific to 3D. This calculator handles 2D dot products.

What does it mean when the dot product is zero?

The vectors are perpendicular (orthogonal). For example, (1, 0) · (0, 1) = 0 because east and north are at right angles. This property is used constantly in physics to decompose forces into independent components and in graphics for lighting calculations.