Vectors in 3D space – sum, dot product, angle
MathematicsVectors & Coordinate GeometryAges 17–18
Use it with your class: students open it from a link or a QR code, with no account needed.
Next in Vectors & Coordinate Geometry
An Oxyz coordinate system rotates with the mouse or a finger, showing two vectors u and v with given coordinates. The simulation draws the coordinate box of each vector and the parallelogram for the sum u + v, and computes lengths, the dot product u·v = x₁x₂ + y₁y₂ + z₁z₂, and the angle between the vectors; drag the coordinate sliders to see when u ⊥ v or when they point the same way. The 2D | 3D switch opens an Oxyz system built with three.js showing u, v, their sum and difference, the coordinate boxes and the angle arc; you can rotate and zoom it, and the numbers stay the same. The Vector product option adds u × v, perpendicular to both by the right-hand rule, the parallelogram and triangle areas |u × v| and ½|u × v|, the plane through O, U and V, physics examples (torque, work) and three self-checking tasks.
Lesson: Vectors in space, coordinate systems, coordinate expressions for vector operations
What it shows
In an Oxyz coordinate system a vector is given by three coordinates, which are the signed lengths of the edges of its coordinate box, and its length is √(x² + y² + z²). The sum u + v is the diagonal of the parallelogram on u and v, and the difference u − v runs from the tip of v to the tip of u. The dot product u·v = x₁x₂ + y₁y₂ + z₁z₂ gives the angle through cos(u, v) = u·v/(|u|·|v|); it is zero exactly when the vectors are perpendicular. The vector product u × v is perpendicular to both, and its length |u||v| sin θ is the parallelogram's area.
How to use
Set the three coordinates of u and v and turn on Coordinate box to link each coordinate to an edge. Drag the background to rotate the view and check that the picture is really three-dimensional; Reset view returns to the start. Adjust one coordinate of v until u·v = 0, then look at the angle from several directions. Switch to 3D for a rotatable, zoomable view with the same numbers. Tick Vector product u × v to see the right-hand rule and the areas.
Parameters you can change
- u: x-coordinate x₁ -5–5
- u: y-coordinate y₁ -5–5
- u: z-coordinate z₁ -5–5
- v: x-coordinate x₂ -5–5
- v: y-coordinate y₂ -5–5
- v: z-coordinate z₂ -5–5
- Vector product u × v
- Area Parallelogram, Triangle
Questions to explore
- With u = (3; 1; 2), find a vector v perpendicular to u. How can you check it without the picture?
- How is the length of a vector calculated from its three coordinates, and does the coordinate box confirm it?
- Two vectors look perpendicular on the screen. Why might they not be, and how do you decide?
