Sure! Let's start by defining the vectors [tex]\( v \)[/tex] and [tex]\( w \)[/tex]:
Vector [tex]\( v = -2i + 3j \)[/tex]
Vector [tex]\( w = 6i - 6j \)[/tex]
First, we need to find the magnitudes of each vector.
1. Magnitude of vector [tex]\( v \)[/tex]:
The formula for the magnitude of a vector [tex]\( ai + bj \)[/tex] is given by:
[tex]\[
\|v\| = \sqrt{a^2 + b^2}
\][/tex]
For [tex]\( v = -2i + 3j \)[/tex]:
[tex]\[
\|v\| = \sqrt{(-2)^2 + 3^2} = \sqrt{4 + 9} = \sqrt{13}
\][/tex]
2. Magnitude of vector [tex]\( w \)[/tex]:
For [tex]\( w = 6i - 6j \)[/tex]:
[tex]\[
\|w\| = \sqrt{6^2 + (-6)^2} = \sqrt{36 + 36} = \sqrt{72} = 6\sqrt{2}
\][/tex]
Now we find the difference in magnitudes [tex]\( \|v\| - \|w\| \)[/tex]:
[tex]\[
\|v\| = \sqrt{13}
\][/tex]
[tex]\[
\|w\| = 6\sqrt{2}
\][/tex]
[tex]\[
\|v\| - \|w\| = \sqrt{13} - 6\sqrt{2}
\][/tex]
Thus, the exact answer is:
[tex]\(
\boxed{\sqrt{13} - 6\sqrt{2}}
\)[/tex]
