Mathematics 1

Geometrical interpretation of the Cross product magnitude - example

• Find the area of triangle with
• vertices: P1(-1,0,1); P2(0,2,2);

P3 (0,-1,2); P2

Solution:

We have to build vectors ,which describe the triangle

=(1,2,1); =(1;-1;1)

=Area of the parallelogram they enclose.

=(3-3)

= ; Area of P1P2P3 =

•  

P1
P2
P3

The Scalar Triple Product

• The volume V of this object is the base area B times the height h = cos.
• B = ,
• points up from the right-hand-rule.
• Hence, we have:

Volume=Bh= =∥a×b∥ ∥c∥ |cosϕ|=
|(a×b)⋅c|.

The scalar product of three vectors a,b and c like the dot product evaluates to a single number.
Base area B
Its absolute value is the volume of the parallelepiped spanned by a,b and c.