Scalar Produkt of two Vektors

The scalar product of two vectors A und B is defined as:

A · B = |A| |B| cos(f_AB)

Here f_AB is the angle between the two vectors. You can also write thsi as the projection of A on B, A||B = |A|·cos(f_AB), multiplied by the length of the vector B:

A · B = (A||B) |B|

Thus the scalar product is the area that one obtains by multiplying the length of B with the component of A parallel to B. The applet above illustrated this. The area is shown in yellow, when the scalar product is positive, and pink when negative.

Das Skalarprodukt ist demnach die Fläche, die man erhält, wenn man die Länge von |B| mit der Komponente von A entlang B multipliziert. Das obige Applet zeigt dies. Die Fläche wird gelb angezeigt, wenn das Skalarprodukt einen positiven Wert hat, und rosa, wenn das Skalarprodukt kleiner als 0 ist.

Please note: you can also project B onto A and get:

A · B = (B||A) |A|

All formulas given here are identical.

Hint: You can grab and drag each vector on its tip and tail to change the scalar product.