if x,y and z are not all zero and connected by the equations `a_(1)x+b_(1)y+c_(1)z=0,a_(z)x+b_(2)y+c_(2)z=0` and `(p_(1)+lambdaq_(1))x+(p_(2)+lambdaq_(2))y+(p_(3)+lambdaq_(3))z=0` show that
`lambda =-|{:(a_(1),,b_(1),,c_(1)),(a_(2) ,,b_(2),,c_(2)),(p_(1) ,, p_(2),,p_(3)):}|-:|{:(a_(1),,b_(1),,c_(1)),(a_(2) ,,b_(2),,c_(2)),(q_(1) ,, q_(2),,q_(3)):}|`

Explanation

To solve the problem, we need to show that the parameter \(\lambda\) can be expressed in terms of the determinants of certain matrices derived from the given equations. Here's a step-by-step solution:

### Step 1: Write down the given equations
We start with the equations:
1. \(a_1 x + b_1 y + c_1 z = 0\)
2. \(a_2 x + b_2 y + c_2 z = 0\)
3. \((p_1 + \lambda q_1)x + (p_2 + \lambda q_2)y + (p_3 + \lambda q_3)z = 0\)

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