Answer:
(D) 2
Step-by-step 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 = [tex]\left[\begin{array}{ccc}1&1&2\\1&p&5\\5&3&4\end{array}\right][/tex]
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
[tex]det(A) = det \left[\begin{array}{ccc}1&1&2\\1&p&5\\5&3&4\end{array}\right] = 0[/tex]
Now, let's calculate the determinant.
[tex]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[/tex]
=> 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
