# Express -i-3j+4k  as a linear combination of vectors 2i+j-4k, 2i-j+3k - Mathematics and Statistics

Express -bari-3barj+4bark   as a linear combination of vectors  2bari+barj-4bark,2bari-barj+3bark

#### Solution

Let bar a = 2hati + hatj - 4hatk , bar b = 2hati -hatj + 3hatk , bar c=3hati+ hatj - 2hatk and r = -hati - 3 hatj + 4hatk

Consider, barr = x bara + ybar b + zbar  c ….(i)

where x, y, z are scalars

-hati - 3 hatj + 4hatk = x(2hati + hatj-4hatk ) + y(2hati-hatj+ 3hatk )+ z(3hati + hatj-2hatk )

-hati - 3 hatj + 4hatk = (2x + 2y + 3z)hati + (x - y + z) hatj + (-4x + 3y - 2z)hatk

By equality of vectors, we get

 2x + 2y + 3z = -1,x - y + z = -3,-4x + 3y -2z = 4

By, Cramer’s rule, we get

D=|[2,2,3],[1,-1,1],[-4,3,-2]|

=2(2-3)-2(-2+4)+3(3-4)

=2(-1)-2(2)+3(-1)

=-2-4-3

=-9≠0

D_x=|[-1,2,3],[-3,-1,1],[4,3,-2]|

=-1(2-3)-2(6-4)+3(-9+4)

=-1(-1)-2(2)+3(-5)

=1-4-15

=-18

D_y=|[2,-1,3],[1,-3,1],[-4,4,-2]|

=2(6-4)+1(-2+4)+3(4-12)

=2(2)+1(2)+3(-8)

=4+2-24

=-18

D_z=|[2,2,-1],[1,-1,-3],[-4,3,4]|

=2(-4+9)-2(4-12)-1(3-4)

=2(5)-2(-8)-1(-1)

=10+16+1

=27

x=D_x/D=-18/-9=2

y=D_y/D=-18/-9=2

z=D_z/D=27/-9=-3

therefore bar r=2 bara+2barb-3barc .

Concept: Vector and Cartesian Equations of a Line - Linear Combination of Vectors
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2014-2015 (October)

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