To find [tex]\(\| v \| - \| w \|\)[/tex] for the vectors [tex]\(v = -6\mathbf{i} + 4\mathbf{j}\)[/tex] and [tex]\(w = -3\mathbf{i} + 2\mathbf{j}\)[/tex], we need to follow these steps:
1. Find the magnitude (norm) of [tex]\(v\)[/tex]:
The norm of a vector [tex]\(v = a\mathbf{i} + b\mathbf{j}\)[/tex] is given by:
[tex]\[
\| v \| = \sqrt{a^2 + b^2}
\][/tex]
For [tex]\(v = -6\mathbf{i} + 4\mathbf{j}\)[/tex]:
[tex]\[
\| v \| = \sqrt{(-6)^2 + 4^2}
\][/tex]
[tex]\[
\| v \| = \sqrt{36 + 16}
\][/tex]
[tex]\[
\| v \| = \sqrt{52}
\][/tex]
[tex]\[
\| v \| = \sqrt{4 \cdot 13}
\][/tex]
[tex]\[
\| v \| = 2\sqrt{13}
\][/tex]
2. Find the magnitude (norm) of [tex]\(w\)[/tex]:
Again, using the formula for the norm of a vector for [tex]\(w = -3\mathbf{i} + 2\mathbf{j}\)[/tex]:
[tex]\[
\| w \| = \sqrt{(-3)^2 + 2^2}
\][/tex]
[tex]\[
\| w \| = \sqrt{9 + 4}
\][/tex]
[tex]\[
\| w \| = \sqrt{13}
\][/tex]
3. Calculate the difference between the magnitudes:
[tex]\[
\| v \| - \| w \| = 2\sqrt{13} - \sqrt{13}
\][/tex]
Factor out [tex]\(\sqrt{13}\)[/tex]:
[tex]\[
\| v \| - \| w \| = \sqrt{13}(2 - 1)
\][/tex]
Simplify:
[tex]\[
\| v \| - \| w \| = \sqrt{13}
\][/tex]
Putting it all together:
[tex]\[
\| v \| - \| w \| = \sqrt{13}
\][/tex]
Thus, the exact answer is:
[tex]\[
\sqrt{13}
\][/tex]
