Dot, cross and scalar triple products in 3D

Direct answer

Build three vectors in 3D and see the dot product, the cross product as an area vector and the scalar triple product as a volume.

a × b (yellow) is perpendicular to both a and b, and its length is the shaded parallelogram's area. The green box built on a, b and c has volume |[a b c]|.

Simulation

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Dot, Cross and Box Product of Vectors in 3D

Vectors

a: x 2

a: y 0

a: z 0

b: x 1

b: y 2

b: z 0

c: x 0

c: y 1

c: z 2

Drag to rotate, two fingers or right-drag to pan, scroll or pinch to zoom. The toolbar has views, cutaway, measure and “show the maths”.

What's happening? a × b (yellow) is perpendicular to both a and b, and its length is the shaded parallelogram's area. The green box built on a, b and c has volume |[a b c]|.

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Where it is in NCERT: Class 12 Mathematics Part II, “Vector Algebra” (Types of Vectors, Addition of Vectors, Multiplication of a Vector by a Scalar, Product of Two Vectors) · ncert.nic.in
Topic notes and formulas: Vector Algebra
Printable formula sheet
Chapter-wise JEE Main PYQs (10)