How to Determine Perpendicular and Collinear Vectors with 2 Variables

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To determine if vectors a = (2, p, 8) and b = (q, 4, 12) are perpendicular, the dot product must equal zero, leading to the equation 2q + 4p + 96 = 0. The values for p and q that satisfy this condition are p = 1 and q = -50. For collinearity, the vectors must be scalar multiples of each other, which results in the equations 2/q = p/4 = 8/12. Solving these gives p = 8/3 and q = 3. Understanding the definitions and relationships between the vectors is crucial for solving both parts of the problem.
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Homework Statement


This problem actually has 2 parts.
For vectors a= (2,p,8) and b= (q,4,12), determine the values of p and q so that the vectors are a)perpendicular b) collinear

textbook answer: a) p= 1 and q= -50 (answers may vary) b) p= 8/3 and q=3

Homework Equations


a*b = 0

a*b = a1b1 + a2b2 + a3b3

3. The Attempt at a Solution
So for the first part, I used a*b = 0 and I don't know how to do the second part

a*b = 0
(2,p,8)*(q,4,12) = 0
2q + 4p + 96 =0

I tried isolating p and q and substituting them but I end up with 0.
 
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First part: What do you mean by "end up with zero"? You are supposed to pick p and q so that the inner product is zero.

Second part: What does it mean for two vectors to be collinear?
 

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