1. Condition 1: Angle between a and b is acute.
If the angle θ is acute, then \cos \theta > 0, which implies \mathbf{a} \cdot \mathbf{b} > 0.
\mathbf{a} \cdot \mathbf{b} = (x)(2x) + (-3)(x) + (-1)(-1) > 0
Factoring the quadratic:
This inequality holds when:
x < 1/2 \quad \text{or} \quad x > 1
2. Condition 2: Angle between b and positive y-axis is between π/2 and π.
The positive direction of the y-axis is represented by the unit vector j.
If the angle ϕ is between π/2 and π, then \cos \phi < 0.
\mathbf{b} \cdot \mathbf{j} < 0
(2x \mathbf{i} + x \mathbf{j} - \mathbf{k}) \cdot \mathbf{j} < 0
3. Combining both conditions:
We need to find the intersection of:
Condition 1: x∈(−∞,1/2)∪(1,∞)
Condition 2: x∈(−∞,0)
The intersection of these two sets is:
Correct Option: (c) \{x : x < 0\}
