Unit 4 - Activity 9

ACTIVITY 4.9
BASES NOT CANONICAL

In the right figure you have the first example of a not canonical base ; it is formed by two vectors not null neither parallel

and

(they are also unit and perpendicular, like

and

, but we will talk about that in the next activity).

The components of another vector in this base are the two scalars x and y that they let write like a linear combination of and : = x+ y.

Like this, in the case of the figure, vector can express in in two ways:
= 1,3+ 4
= 3.3+ 2.6
therefore, while the components of in the canonical base , are (1.3 , 4), components of in the base not canonical , are (3.3 , 2.6).

To pass to the expression of in the base , to the expression of in the base , we have to know how to write vectors and like a linear combination of the vectors and (or how to express and like a linear combination of and); they are called formulas of changing of base. In the homework we will use them.

INTERACTIVE ACTIVITY

In the applet of the right side you have two bases of the vectors of the plane (both formed by unit and perpendicular vectors ), the and , and the and .

a) Draw vector =-+5 and you observe its components of the base and .
b) Draw vector =-3+ and observe its components of the base and .

Vary the base and (moving the point P) until the angle xx' being of -18.35º. Then:
c) Draw vector =-+5 and observe its components in the base and .
d) Draw vector =-3+ and observe its components in the base and .

SOLUTION

HOMEWORK

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

We consider two bases of vectors of the plane:
- the canonical and
- another formed by two vectors and , which we know that = 0.8+ 0.6 and = -0.6+ 0.8
1) If components of a vector in the base and are (2,5), calculate components of in the base and .
2) If components of a vector in the base and are (6,3), calculate components of in the base and .

3) Do a graphic where there are represented the two bases an the two vectors and .

END OF ACTIVITY 4.9
BASES NOT CANONICAL

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