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Question
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"`.
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" + "y" - "z")hat"i" + (- "x" - 2"y" + 3"z")hat"j" + ("3x" + "4y" - "5z")hat"k"`
By equality of vectors,
2x + y - z = 1
- x - 2y + 3z = 4
3x + 4y - 5z = - 4
We have to solve these equations by using Cramer’s Rule.
D = `|(2,1,-1),(-1,-2,3),(3,4,-5)|`
= 2 (-2) + (-1) (-4) + (-1) (2)
= - 4 + 4 - 2
= -2
Dx = `|(1,1,-1),(4,-2,3),(-4,4,-5)|`
= (1) (-2) + (-1) (-8) + (-1) (8)
= - 2 + 8 - 8
= -2
Dy = `|(2,1,-1),(-1,4,3),(3,- 4,-5)|`
= 2 (-8) - 1 (-4) - 1 (-8)
= - 16 + 4 + 8
= - 4
Dz = `|(2,1,1),(-1,-2,4),(3,4,-4)|`
= 2 (-8) - 1 (-8) + (1) (2)
= - 16 + 8 + 2
= - 6
∴ x = `"D"_"x"/"D" = (-2)/(-2) = 1`
∴ y = `"D"_"y"/"D" = (- 4)/(-2) = 2`
∴ z = `"D"_"z"/"D" = (-6)/(-2) = 3`
∴ `bar"p" = bar"a" + 2bar"b" + 3bar"c"`
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