Question

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

 

 

Answer 1

`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 .`