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Vector Math guide

The Vector Math Calculator performs 2D and 3D vector operations, including addition, dot product, cross product, and magnitude. It visualizes the vectors on a graph and provides detailed analysis like the angle between them and their projections.

Worked examples

Calculate the dot product of two vectors

  1. Select 2D mode and the "Dot Product" tab.
  2. Set vector u to (3, 4) and vector v to (1, 2).
  3. Look at the Result card and Analysis section.
Result: u · v = 11

Multiply matching components and add: (3×1) + (4×2) = 3 + 8 = 11. A positive dot product means the angle is acute.

Calculate the cross product in 3D

  1. Select 3D mode and the "Cross Product" tab.
  2. Set vector u to (1, 0, 0) and vector v to (0, 1, 0).
  3. Click Calculate.
Result: u × v = (0, 0, 1)

The cross product of the x-axis and y-axis produces the z-axis. It is perpendicular to both original vectors.

Find the angle between two vectors

  1. Select the "Angle" tab.
  2. Set vector u to (1, 0) and vector v to (1, 1).
  3. Check the angle result.
Result: θ = 45°

Using cos(θ) = (u·v) / (|u||v|). The dot product is 1, |u| is 1, and |v| is √2. cos(θ) = 1/√2, which is 45°.

Tips

  • •The dot product gives a scalar (a single number), while the cross product gives a new vector.
  • •The cross product is only defined for 3D vectors (and 7D, but this tool handles 3D).
  • •A dot product of 0 means the vectors are exactly perpendicular (orthogonal).

Frequently asked questions

What is a vector?
A mathematical object that has both magnitude (length) and direction, commonly used to represent forces, velocities, or displacements.
What is the difference between a vector and a scalar?
A scalar only has magnitude (like temperature or mass), while a vector has both magnitude and direction.
How do you add two vectors?
Add their corresponding components. Geometrically, it is done by placing them head-to-tail to form a parallelogram.
What is the dot product?
Also called the scalar product, it measures how much two vectors point in the same direction. It is calculated as u·v = |u||v|cos(θ).
What does it mean if the dot product is zero?
It means the vectors are orthogonal (perpendicular) to each other.
What is the cross product?
A vector operation (3D only) that produces a third vector perpendicular to both original vectors. It is calculated using a determinant matrix.
What is the right-hand rule?
A rule used to determine the direction of the cross product. If you curl your right fingers from vector u to v, your thumb points in the direction of u × v.
What is the magnitude of a vector?
The length of the vector, calculated using the Pythagorean theorem: |u| = √(x² + y²) for 2D, or √(x² + y² + z²) for 3D.
What is a unit vector?
A vector with a magnitude of exactly 1. It is used to indicate direction only. You find it by dividing a vector by its magnitude.
What is vector projection?
The "shadow" of one vector cast onto another. It decomposes a vector into two parts: one parallel to the second vector and one perpendicular.
Can you multiply three vectors together?
Yes, using the scalar triple product (u · (v × w)), which gives the volume of the parallelepiped formed by the three vectors.
What is the triangle inequality for vectors?
The magnitude of the sum of two vectors is always less than or equal to the sum of their magnitudes: |u + v| ≤ |u| + |v|.
How are vectors used in physics?
They represent any quantity with direction: force, velocity, acceleration, electric fields, and magnetic fields.
How are vectors used in computer graphics?
Vectors determine surface normals (which way a face is pointing) for lighting calculations, and camera movement.
What happens if I try to cross-product in 2D?
The mathematical cross product requires a third dimension. The calculator requires you to switch to 3D mode for this operation.
Is vector addition commutative?
Yes. u + v = v + u.
Is the cross product commutative?
No. u × v = -(v × u). Reversing the order reverses the direction of the resulting vector.
What are the standard basis vectors?
They are the unit vectors pointing along the x, y, and z axes, usually denoted as î, ĵ, and k̂ (or x̂, ŷ, ẑ).
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