3D vectors and planes — LearnRoost Playground

Examples:

Show a × b & area parallelogram Show plane through P₀ & normal n Show 3D coordinate drop-lines

Vector a = (3, 1, 2)

a_x 3

a_y 1

a_z 2

Vector b = (-1, 3, 1)

b_x -1

b_y 3

b_z 1

Point on plane P₀ = (1, 1, 1)

x₀ 1

y₀ 1

z₀ 1

Plane normal vector n

Use n = a × b Custom n

Currently using n = a × b so the plane is parallel to both a and b.

1. Scalar (dot) product & angle θ

Acute angle

The dot product measures how much a and b point in the same direction. It is positive when θ < 90°, zero when perpendicular (θ = 90°), and negative when θ > 90°.

2. Vector (cross) product a × b

Perpendicular to a and b

The cross product gives a 3D vector perpendicular to both a and b (by the right-hand rule). Its magnitude |a × b| equals the area of the shaded parallelogram spanned by a and b.

3. Plane through P₀ with normal n

r · n = d

Any point r = (x, y, z) lies on the plane if the displacement vector (r − p₀) from P₀ is at 90° to the normal vector n, so (r − p₀) · n = 0.