3D Vector Visualizer

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Visualize and calculate 3D vectors interactively. See vector addition, cross product, dot product, projections, and angles between vectors rendered as 3D arrows with real-time calculations.

Results

|A|2.2361
|B|2.2361
A \u00B7 B2.0000
A \u00D7 B(1.0000, -2.0000, 4.0000)
|A \u00D7 B|4.5826
Angle66.4218°
Angle (rad)1.1593
proj_B(A)(0.0000, 0.8000, 0.4000)
|proj|0.8944
\u00C2 (unit A)(0.8944, 0.4472, 0.0000)
B\u0302 (unit B)(0.0000, 0.8944, 0.4472)
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Understanding 3D Vectors and Vector Operations

Vectors are fundamental mathematical objects that represent both magnitude and direction in space. In three dimensions, a vector is described by three components (x, y, z) and can be visualized as an arrow pointing from the origin to a point in 3D space.

Key Vector Operations

  • Magnitude: The length of a vector, calculated as |v| = sqrt(x² + y² + z²). It represents the distance from the origin to the vector's tip.
  • Dot Product: A · B = AxBx + AyBy + AzBz. This scalar quantity measures how much two vectors align. When zero, the vectors are perpendicular.
  • Cross Product: A × B produces a new vector perpendicular to both input vectors. Its magnitude equals the area of the parallelogram formed by A and B.
  • Angle Between Vectors: Found using cos(θ) = (A · B) / (|A| × |B|). This gives the smallest angle between two vectors.

Vector Projection

The projection of vector A onto vector B gives the component of A in the direction of B. It is calculated as proj_B(A) = ((A · B) / (B · B)) × B. This is widely used in physics to decompose forces and in computer graphics for lighting calculations.

Applications

3D vectors are essential in physics (forces, velocities, fields), computer graphics (normals, lighting, transformations), robotics (kinematics), and game development (movement, collision detection). Understanding vector operations is a cornerstone of linear algebra and applied mathematics.

Frequently Asked Questions

What is the cross product of two vectors?
The cross product A × B is a vector operation that produces a new vector perpendicular to both input vectors. Its magnitude equals |A| × |B| × sin(θ), where θ is the angle between them. The direction follows the right-hand rule: point your fingers from A toward B, and your thumb points in the direction of A × B.
What is the difference between dot product and cross product?
The dot product produces a scalar (single number) that measures alignment between vectors, while the cross product produces a new vector perpendicular to both inputs. The dot product is commutative (A · B = B · A), but the cross product is anti-commutative (A × B = -(B × A)).
How do I find the angle between two vectors?
The angle θ between two vectors A and B is found using the formula cos(θ) = (A · B) / (|A| × |B|). First compute the dot product and both magnitudes, then take the inverse cosine (arccos) of the result. The angle will be between 0° and 180°.
What does vector projection mean?
The projection of vector A onto vector B is the component of A that lies along the direction of B. Geometrically, it is the shadow of A cast onto B. The formula is proj_B(A) = ((A · B) / |B|²) × B. It is widely used in physics to resolve forces into components.
Why is the cross product zero for parallel vectors?
When two vectors are parallel, the angle between them is 0° or 180°, making sin(θ) = 0. Since the cross product magnitude is |A| × |B| × sin(θ), it becomes zero. Geometrically, parallel vectors do not span a parallelogram with area, so there is no meaningful perpendicular direction.