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Express ijki^+4j^-4k^ as the linear combination of the vectors ijkijk2i^-j^+3k^,i^-2j^+4k^ and ijk-i^+3j^-5k^. - Mathematics and Statistics

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Sum

Express `hat"i" + 4hat"j" - 4hat"k"` as the linear combination of the vectors `2hat"i" - hat"j" + 3hat"k", hat"i" - 2hat"j" + 4hat"k"` and `- hat"i" + 3hat"j" - 5hat"k"`.

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Solution

Let `bar"a" = 2hat"i" - hat"j" + 3hat"k"`, 
`bar"b" = hat"i" - 2hat"j" + 4hat"k"`, 
`bar"c" = - hat"i" + 3hat"j" - 5hat"k"`
`bar"p" = hat"i" + 4hat"j" - 4hat"k"`

Suppose `bar"p" = "x"bar"a" + "y"bar"b" + "z"bar"c"`.

Then, `hat"i" + 4hat"j" - 4hat"k" = "x"(2hat"i" - hat"j" + 3hat"k") + "y"(hat"i" - 2hat"j" + 4hat"k") + "z"(- hat"i" + 3hat"j" - 5hat"k")`

∴ `hat"i" + 4hat"j" - 4hat"k" = (2"x" + 2"y" - "z")hat"i" + (- "x" - 2"y" + 3"z")hat"j" + ("3x" + "4y" - "5z")hat"k"`

By equality of vectors,

2x + 2y - z = 1

- x - 2y + 3z = 4

3x + 4y - 5z = - 4

We have to solve these equations by using Cramer’s Rule.

D = `|(2,2,-1),(-1,-2,3),(3,4,-5)|`

= 2(10 - 12) - 2(5 - 9) - 1 (-4 + 6)

= - 4 + 8 - 2

= 2 ≠ 0

Dx = `|(1,2,-1),(4,-2,3),(-4,4,-5)|`

= 1(10 - 12) - 2(- 20 + 12) - 1(16 - 8)

= - 2 + 16 - 8

= 6

Dy = `|(2,1,-1),(-1,4,3),(3,- 4,-5)|`

= 2(- 20 + 12) - 1(5 - 9) - 1(4 - 12)

= - 16 - 4 - 8

= - 28

Dz = `|(2,2,1),(-1,-2,4),(3,4,-4)|`

= 2(8 - 16) - 2(4 - 12) + 1(- 4 + 6)

= - 16 - 16 + 2

= - 30

∴ x = `"D"_"x"/"D" = 6/2 = 3`

∴ y = `"D"_"y"/"D" = (- 28)/2 = - 14`

∴ z = `"D"_"z"/"D" = (-30)/2 = - 15`

∴ `bar"p" = 3bar"a" - 14bar"b" - 3bar"c"`

Concept: Vectors and Their Types
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