The number of integral values of p for which the vector ( p +1) i -3 j + p k , p i +( p +1) j -3 k and -3 i + p j +( p +1) k are linearly dependent vectors, are

Step-by-step solution

The vectors are linearly dependent when their scalar triple product vanishes, which occurs when the determinant of their component matrix equals zero. vmatrix p+1 & -3 & p p & p+1 & -3 -3 & p & p+1 vmatrix = 0 Expanding the determinant yields: (p+1)[(p+1)^2 + 3p] + 3[p(p+1) - 9] + p[p^2 + 3(p+1)] = 0 Simplifying the expression: (p+1)(p^2 + 5p + 1) + 3p^2 + 3p - 27 + p^3 + 3p^2 + 3p = 0 Combining like terms produces the cubic equation: 2p^3 + 12p^2 + 12p - 26 = 0 Dividing through by 2: p^3 + 6p^2 + 6p - 13 = 0 Testi