3D Vector Decomposition

u

1

-7

u

2

0

u

3

-8

v

1

0

v

2

-9

v

3

0

w

1

0

w

2

0

w

3

-7

x

1

1

x

2

1

x

3

1

show axis

c

1

u

+

c

2

v

+

c

3

w

=

x

System's Augmented Matrix =

-7	0	0	1
0	-9	0	1
-8	0	-7	1

c

1

= -

1

7

,

c

2

= -

1

9

,

c

3

=

1

49

This Demonstration shows the decomposition of a vector in 3D. The components of the three vectors



u

,



v

, and

w

defining the directions of decomposition and of the vector



x

to be decomposed are adjusted using the sliders. The scalar multipliers

c

1

,

c

2

,

c

3

of each of the three vectors are calculated and the prism defining the decomposition is graphed.

The scalar multipliers

c

1

,

c

2

,

c

3

are obtained by solving a linear system having an augmented matrix with columns defined by the components of



u

,



v

,

w

, and



x

. If the determinant of the system is zero, corresponding to



u

,



v

, and

w

being linearly dependent, a message is displayed prompting the user to make another selection.