ACTIVITY 5.4
PROPERTIES OF THE SCALAR PRODUCT- 2

The scalar product of two vectors is distributive with respect to the sum of vectors:

·(+) = · + ·

To demonstrate it, we take into account (superior figure in the right side):

1) Firstly, we can calculate ·(+) making the product of|| for the projection of + on : ·(+) = ||(+)_|u

2) However, you can see in the figure, the projection of a sum of vectors on another vector, is the same that the sum of projections:
                                    (+ )_|u = _|u + _|u

3) Linking the two previous identities , we have:

·(+)=||(+)_|u =||(_|u+_|u)=||_|u+||_|u=·+·

It can object that we have prepared the upper figure on the right side very well to make the affirmation 2). What does it happen if the projection on of any of the vectors , or + have opposite direction to the vector (just as pass, for instance, in the lower figure)? We remember that these projections become with sign, therefore, some projections can be negative. What does it happen, then? To answer this question it has prepared itself next interactive activity .

INTERACTIVE ACTIVITY

The figure of the right side manifests the distributivity ·(+)=·+·

Check that also is verified this distributivity when:

1) The projection of onhas opposite direction to direction of .

2) The projection ofonhas opposite direction to direction of.

3) Two previous directions (ofand ofon) have opposite direction to direction of.

SOLUTION

END OF ACTIVITY 5.4
PROPERTIES OF THE SCALAR PRODUCT- 2