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Home > Matrix & Vector calculators > Decomposition of vector in basis example
20. Decomposition of vector in basis example ( Enter your problem )
( Enter your problem )
  1. Example-1
  2. Example-2
 		Other related methods
  1. Addition/Subtraction of two vectors
  2. Scalar Multiplication of vectors
  3. Dot Product of two vectors
  4. Cross Product of two vectors
  5. Magnitude(length) of a vector
  6. Unit vector
  7. Direction cosines of a vector
  8. Component form of a vector given two points
  9. Angle between two vectors
  10. Vector projections of b onto a
  11. Orthogonal vectors
  12. Collinear vectors
  13. Coplanar vectors
  14. Scalar triple product
  15. Vector triple product
  16. Area of triangle determined by two vectors
  17. Area of parallelogram determined by two vectors
  18. Volume of pyramid determined by vectors
  19. Volume of Parallelepiped determined by vectors
  20. Decomposition of vector in basis
  21. Linearly dependent and linearly independent vectors
1. Example-1
(Previous example)		21. Linearly dependent and linearly independent vectors
(Next method)

2. Example-2


1. Find Decompose(A,B,C)
`A=(3,7)`,`B=(-1,2)`,`C=(3,20)`

Solution:
Here `vec A=(3,7),vec B=(-1,2),vec C=(3,20)`


Form equation from vectors
`vec C = x vec A + y vec B`

So system of linear equations are
`3x-1y=3`

`7x+2y=20`

Solution of equations using Elimination method

Total Equations are `2`

`3x-y=3 -> (1)`

`7x+2y=20 -> (2)`


Select the equations `(1)` and `(2)`, and eliminate the variable `y`.

`3x-y=3`` xx 2->````6x``-``2y``=``6```
+
`7x+2y=20`` xx 1->````7x``+``2y``=``20```
```13x``=``26`` -> (3)`



Now use back substitution method
From (3)
`13x=26`

`=>x=(26)/(13)=2`

From (1)
`3x-y=3`

`=>3(2)-y=3`

`=>-y+6=3`

`=>-y=3-6=-3`

`=>y=3`

Solution using Elimination method.
`x=2,y=3`


`x=2,y=3`

So, `vec C=2vec (A) +3vec (B)` 2. Find DECOMPOSE(B,C,A)
`A=(1,2)`,`B=(3,1)`,`C=(8,1)`

Solution:
Here `vec A=(1,2),vec B=(3,1),vec C=(8,1)`


Form equation from vectors
`vec A = x vec B + y vec C`

So system of linear equations are
`3x+8y=1`

`x+y=2`

Solution of equations using Elimination method

Total Equations are `2`

`3x+8y=1 -> (1)`

`x+y=2 -> (2)`


Select the equations `(1)` and `(2)`, and eliminate the variable `x`.

`3x+8y=1`` xx 1->````3x``+``8y``=``1```
−
`x+y=2`` xx 3->````3x``+``3y``=``6```
```5y``=``-5`` -> (3)`



Now use back substitution method
From (3)
`5y=-5`

`=>y=(-5)/(5)=-1`

From (2)
`x+y=2`

`=>x+(-1)=2`

`=>x-1=2`

`=>x=2+1=3`

Solution using Elimination method.
`x=3,y=-1`


`x=3,y=-1`

So, `vec A=3vec (B) -vec (C)`

This material is intended as a summary. Use your textbook for detail explanation.
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1. Example-1
(Previous example)		21. Linearly dependent and linearly independent vectors
(Next method)


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