Question

How to check whether the vector 4i+13j-18k is a linear combination of the vectors i-2j+3k & 2i+3j-4k .

Answer 1

Solution:

If Vector A=4i+13j-18k is a linear combination of the vectors, B= i-2j+3k & C=2i+3j-4k , then

A=u B + m C

4 i+13 j-18 k= u( i -2 j + 3 k)+ m(2 i + 3 j -4 k)

4 i + 13 j - 18 k = (u +2 m)i +(-2 u+3 m)j+(3 u-4 m)k

Equating LHS and RHS

u + 2 m = 4--------(1)

- 2 u + 3 m= 13-------(2)

3 u - 4 m = -18------------(3)

Solving 1 and 2

2 equation (1) +equation 2

2 u + 4 m - 2 u + 3 m =8 +13

7 m=21

m=3

Substituting the value of m in equation 1

u +6=4

u=4-6

u=-2

Putting the value of u and m in equation (3)

LHS= 3 × (-2) -4×(3)

=-6 -12

=-18

=RHS

Hence we can say that,  vector 4i+13j-18k is a linear combination of the vectors i-2j+3k & 2i+3j-4k .