To determine the slope of line \( b \) given that lines \( a \) and \( b \) are perpendicular and knowing the slope of line \( a \) is 3, we need to understand the relationship between the slopes of two perpendicular lines.
The slopes of two perpendicular lines are related through the following rule:
[tex]\[ \text{slope of line } a \times \text{slope of line } b = -1 \][/tex]
Given:
[tex]\[ \text{slope of line } a = 3 \][/tex]
We can substitute this value into the equation to find the slope of line \( b \):
[tex]\[ 3 \times \text{slope of line } b = -1 \][/tex]
To isolate the slope of line \( b \), we solve for it by dividing both sides of the equation by 3:
[tex]\[ \text{slope of line } b = \frac{-1}{3} \][/tex]
Thus, the slope of line \( b \) is:
[tex]\[ -\frac{1}{3} \][/tex]
Hence, the correct answer is:
[tex]\[ \boxed{ -\frac{1}{3} } \][/tex]
