For which value of k the vector u=[[1],[-2],[k]] is a

Question

For which value of

k

the vector

u = [\begin{matrix} 1 \\ - 2 \\ k \end{matrix}]

is a linear combination of the vectors

v_{1} = [\begin{matrix} 3 \\ 0 \\ - 2 \end{matrix}]

and

v_{2} = [\begin{matrix} 2 \\ - 1 \\ - 5 \end{matrix}]

.

Answer 1

put the given values in the equation

u = α v_{1} + β v_{2}

[\begin{matrix} 1 \\ - 2 \\ k \end{matrix}] = α [\begin{matrix} 3 \\ 0 \\ - 2 \end{matrix}] β [\begin{matrix} 2 \\ - 1 \\ - 5 \end{matrix}]

Now solve the given system get

3 α + 2 β = 1

...(i)

0 α - β = - 2

...(ii)

- 2 α - 5 β = k

...(iii)
Now find the value of

α, β

using equations i and ii
From equation (ii)

- β = - 2

β = 2

Put this value in equation i and solve for

α

3 α + 2 (2) = 1

3 α + 4 = 1

3 α = - 3

α = - 1

Now put the value of

α, β

in equation (iii) and solve it for k

- 2 (- 1) - 5 (2) = k

2 - 10 = k

k = - 8

Answer:
Value of k is -8

Expert Answer