Unit 5 - Activity 5

ACTIVITY 5.5
OBTAINING THE SCALAR PRODUCT FROM THE COMPONENTS

We remember that the two vectors =(1,0) and =(0,1) form a base of the vectors of the plane. Now, we try to calculate the scalar product of two vectors and knowing their components in the base and (instead of their modules and the angle that they form):

= (a₁,a₂) = a₁+ a₂
= (b₁,b₂) = b₁+ b₂

Using the viewed properties to the previous activities, we can write:

·= (a₁+ a₂)·(b₁+ b₂) = a₁b₁·+ a₁b₂·+ a₂b₁·+ a₂b₂·

Now then, when it is multiplied in a scalar way the vectors and , verifies

· = · = 1
· = · = 0

Substituting the expression of the scalar product of and , we get an important result that it let us calculate directly the scalar product of two vectors knowing their components:

·= (a₁,a₂)·(b₁,b₂) = a₁b₁+ a₂b₂

INTERACTIVE ACTIVITY

You have the scalar product of two calculated vectors applying the formula
·= (a₁,a₂)·(b₁,b₂) = a₁b₁+ a₂b₂

Calculate next scalar products and check that the result using this applet:

1) (4,1)·(2,3)

6) (3,0)·(0,3)

2) (3,-1)·(2,4)

7) (3,2)·(-3,-2)

3) (4,0)·(-2,3)

8) (3,2)·(-2,3)

4) (-2,3)·(1,-2)

9) (-3,-2)·(-2,3)

5) (2,1)·(4,2)

10) (3,2)·(2,-3)

SOLUTION

HOMEWORK

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

Given vectors

=(3,4),

=(1,-2),

=(0,4) and

=(-3,1), calculate next scalars products:

a) ·	b) ·	c) ·	d) ·
e) ·	f ) ·	g) ² =·	h) ²=·
i ) 2·(3+)	j ) (+)·(+)	k) (+)²=(+)·(+)	l ) (+)·(-)

Try to calculate the scalars products i), j), k) and l) in two different ways.

END OF ACTIVITY 5.5
OBTAINING THE SCALAR PRODUCT FROM THE COMPONENTS

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