To determine the slope of line [tex]\( b \)[/tex], we need to use the relationship between the slopes of two perpendicular lines. If two lines are perpendicular, the product of their slopes is [tex]\(-1\)[/tex].
Given that the slope of line [tex]\( a \)[/tex] is [tex]\( -\frac{2}{3} \)[/tex], let's denote the slope of line [tex]\( b \)[/tex] by [tex]\( m_b \)[/tex].
The relationship between the slopes of the two lines can be expressed by the equation:
[tex]\[
m_a \cdot m_b = -1
\][/tex]
Substitute the given slope of line [tex]\( a \)[/tex] into the equation:
[tex]\[
-\frac{2}{3} \cdot m_b = -1
\][/tex]
To solve for [tex]\( m_b \)[/tex], divide both sides of the equation by [tex]\(-\frac{2}{3}\)[/tex]:
[tex]\[
m_b = \frac{-1}{-\frac{2}{3}}
\][/tex]
Simplify the right-hand side of the equation:
[tex]\[
m_b = \frac{-1 \div -1}{\frac{2}{3}} = \frac{1}{\frac{2}{3}} = 1 \cdot \frac{3}{2} = \frac{3}{2}
\][/tex]
Therefore, the slope of line [tex]\( b \)[/tex] is:
[tex]\[
m_b = \frac{3}{2}
\][/tex]
So, the slope of line [tex]\( b \)[/tex] corresponding to the correct answer is:
[tex]\[
\boxed{\frac{3}{2}}
\][/tex]
