Vector Calculator NZ

This calculator performs the most common operations on two three-dimensional vectors: the magnitude of each, the dot product, and the angle between them. Vectors describe quantities with both size and direction, such as velocity, force and displacement, and they are fundamental in physics, engineering, computer graphics, navigation and machine learning. Three operations come up again and again. The magnitude, or length, of a vector is found from its components using Pythagoras in three dimensions. The dot product, which multiplies corresponding components and adds them, measures how much two vectors point in the same direction, and it is zero when they are perpendicular. From the dot product and the magnitudes you can find the angle between two vectors, which is essential for everything from lighting calculations in graphics to working out the work done by a force. You enter the x, y and z components of each vector, and the calculator returns the magnitude of the first vector, the dot product of the two, the angle between them in degrees, and the magnitude of the second vector. The results update live, so you can see how changing a component affects the angle, which helps build intuition. Set the z components to zero to work in two dimensions. Use it to check physics and maths homework, to verify a graphics or engineering calculation, or simply to compute these quantities quickly without slips. The dot product and angle are the building blocks for projections and the cross product too. The calculations are exact for your inputs, with the angle rounded for display.

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5
magnitude of vector a
Dot product12
Angle between53.13°
Magnitude of b4

For 3D vectors; set z to 0 for 2D. The dot product is zero when vectors are perpendicular. Angle is rounded for display.

How it works

The magnitude of a vector is the square root of the sum of its components squared. The dot product multiplies matching components and adds them. The angle between two vectors is the inverse cosine of the dot product divided by the product of the magnitudes, converted to degrees.

Worked example

For vector a = (3, 4, 0) and b = (4, 0, 0), the magnitude of a is the square root of 9 plus 16, which is 5, and the magnitude of b is 4. The dot product is 3 times 4, plus 0, plus 0, equals 12. The angle is the inverse cosine of 12 divided by 5 times 4, which is about 53.13 degrees.

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Frequently asked questions

How do I find the magnitude of a vector?

Take the square root of the sum of the squares of its components. For a 3D vector (x, y, z), the magnitude is the square root of x squared plus y squared plus z squared. This is Pythagoras extended to three dimensions.

What does the dot product tell me?

The dot product measures how much two vectors point the same way. It is positive when they point in similar directions, zero when they are perpendicular, and negative when they oppose. It is used to find angles, projections and the work done by a force.

How do I find the angle between two vectors?

Divide the dot product by the product of the two magnitudes, then take the inverse cosine. The result is the angle between them. This calculator does it for you and reports the angle in degrees.