Unit 4 - Activity 10

ACTIVITY 4.10
BASES NOT CANONICAL AND OBLIQUE

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In the figure of the right side you have an example of not canonical base formed by two vectors

and

that they are united, but they aren't perpendicular; in this case it can say that we have an oblique base.

In this case also, vector can express in two ways:
= 1.3+ 2.9
= 2 + 2
therefore, while the components of in the canonical base , are (1.3 , 2.9), components ofin the not canonical base , are (2,2).

The consideration of these oblique bases surprises us and it seems a searched carefully thing in a world where it is imposed the rectangularity (sheets, squares, doors, windows, PC screens..., all is rectangular). Observe that oblique bases appear simply when we substitute rectangles by parallelograms.

INTERACTIVE ACTIVITY

In the applet of the right side you have two bases of vectors of the plane : canonical and , and the and , of united vectors , but not perpendicular.

a) Draw vector=3+4 and observe its components in the base and .
b) Draw vector=-2+2 and see its components in the base and .

Vary the base and (moving points P and Q) until the angle xx' being of 15º, and the angle x'y' of 120º . Then:

c) Draw vector=4+3 and see its components in the base and .
d) Draw vector=4- and see its components in the base and .

HOMEWORK

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

The most general case of a not canonical base arises when it is formed by two vectors that they aren't neither unit not perpendicular (but you remember that they have to be not parallel and none of them null). For instance, the two vectors

and

of the figure. You observe that

= 2

+

and

= -

+

, and then you answer :
1) If components of a vector

in the base

and

are (3,-3), calculate components of

in the base

and

.
2) If components of a vector

in the base

and

are (-2,3), calculate components of

in the base

and

.
3) Represent two bases and two vectors

and

in the same graphic.

END OF ACTIVITY 4.10
BASES NOT CANONICAL AND OBLIQUE

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