Question

Consider the following vectors in R3:u=(1,1,3) , v=(2,3,k) and w=(k,6,14)Based on the information above, determine, if it exists, the value or values of the parameter k such that the vectors u, v, and w are linearly independent.

Answer 1

Remember that

You can verify if a set of vectors is linearly independent by computing the determinant of a matrix whose columns are the vectors you want to check.

They are linearly independent if, and only if, this determinant is not equal to zero.

so

we have

$\begin{bmatrix}{1} & {2} & {k} \\ {1} & {3} & {6} \\ {3} & {k} & {14}\end{bmatrix}$

The determinant of the given matrix is calculated as

$Det=(1)*det\begin{bmatrix}{3} & {6} \\ {k} & {14}\end{bmatrix}-(2)*det\begin{bmatrix}{1} & {6} \\ {3} & {14}\end{bmatrix}+(k)*det\begin{bmatrix}{1} & {3} \\ {3} & {k}\end{bmatrix}$$Det=(1)*[14*3-6*k]-(2)*[14*1-6*3]+(k)*[k*1-3*3]$$\begin{gathered} Det=(1)*[42-6k]-(2)*[14-18]+(k)*[k-9] \\ Det=42-6k+8+k^2-9k \\ Det=k^2-15k+50 \end{gathered}$

Equate to zero the quadratic equation

$k^2-15k+50=0$

Solving by the formula

a=1

b=-15

c=50

$k=\frac{-(-15)\pm\sqrt{-15^2-4(1)(50)}}{2(1)}$$k=\frac{15\pm5}{2}$

The values of k are

k=10 and k=5

Therefore

The values of k are 5 and 10