Question

For 15 points!!!

If vectors i+j+2k, i+pj+5k and 5i+3j+4k are linearly dependent, the value of p is what?
Please I'm looking for the explanation. i already know the answer is 2, i just don't know how.​

Answer 1

(D) 2

Explanation:

If two or more vectors are linearly dependent, then one of them can be expressed as a linear combination of the rest. In other words, vectors are dependent linearly on one another if the determinant of the matrix that they form is zero.

Given vectors:

i+j+2k

i+pj+5k

5i+3j+4k

Now, since they are linearly dependent, then the determinant of their matrix should be zero. i.e

Let their matrix be A and is given by;

A =
`\left[\begin{array}{ccc}1&1&2\\1&p&5\\5&3&4\end{array}\right]`

Where;

the first column holds the i components of the vectors

the second column holds the j components of the vectors and

the third column holds the k components of the vectors

=> |A| = det(A) = 0

`det(A) = det \left[\begin{array}{ccc}1&1&2\\1&p&5\\5&3&4\end{array}\right] = 0`

Now, let's calculate the determinant.

`det \left[\begin{array}{ccc}1&1&2\\1&p&5\\5&3&4\end{array}\right] = 1(4p - 15) - 1(4 - 25) + 2(3 - 5p) = 0`

=> 4p -15 - 4 + 25 + 6 - 10p = 0

=> 4p - 10p + 12 = 0

=> 6p = 12

=> p = 2

Therefore, the value of p for which the vectors are linearly dependent is 2