ACTIVITY 5.2
GEOMETRICAL INTERPRETATION OF THE SCALAR PRODUCT

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We remember the definition of scalar product of two vectors and :

·= ||||cos()

We observe that  ||cos()  is the orthogonal projection of vector onto vector . If we express this projection like this|a , we can write

·= |||a

In a similar way, considering that  ||cos() =|b  is the orthogonal projection of onto , it is arrived to the conclusion that we can write

·= |||b

Therefore, we have a geometrical interpretation of scalar product of two vectors (and another form to calculate it): is the product of the module of one of them for the orthogonal projection of the other on it.

This projection takes sign. That is to say, if direction of this projection is opposite to the first vector, this projection is negative.

INTERACTIVE ACTIVITY

You have two vectors and and its scalar product ·= 27. Also you have the projection of on .

1) Moving point C try to get other vectors that multiplied in a scalar way by continues giving 27. Where are located the extremes C of these vectors?

2) Also moving C try to get vectors that verify ·= - 27.

3) Finally, try to get vectors that verify
·= 0 . Which position do these vectors have with respect to vector ?

SOLUTION

HOMEWORK
(Do you want a Word document prepared for making this work?)


Draw next pair of vectors and (to make uniform a little the results, draw always horizontal vector and with right direction and units in centimetres):

a) || = 6, || = 4
    ·= 18

b) || = 4, || = 5
    ·= 4

c) || = 3, || = 4
    ·= - 9

d) || = 5, || = 3
    ·= 15

e) || = 4, || = 3
    ·= -12

f ) || = 5, || = 2
    ·= 0

END OF ACTIVITY 5.2
GEOMETRICAL INTERPRETATION OF THE SCALAR PRODUCT

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