Given two vectors [tex]\(\vec{a}\)[/tex] and [tex]\(\vec{b}\)[/tex], we are told that:
1. [tex]\(\vec{a} + \vec{b}\)[/tex] is perpendicular to [tex]\(\vec{a}\)[/tex].
2. [tex]\(2\vec{a} + \vec{b}\)[/tex] is perpendicular to [tex]\(\vec{b}\)[/tex].
We need to find the ratio [tex]\(\frac{|\vec{b}|}{|\vec{a}|}\)[/tex].
1. Step 1: Express the perpendicularity condition for [tex]\(\vec{a} + \vec{b}\)[/tex] with [tex]\(\vec{a}\)[/tex].
If two vectors are perpendicular, their dot product is zero. Therefore,
[tex]\[
(\vec{a} + \vec{b}) \cdot \vec{a} = 0
\][/tex]
Expanding this using the distributive property of the dot product:
[tex]\[
\vec{a} \cdot \vec{a} + \vec{b} \cdot \vec{a} = 0
\][/tex]
We know that [tex]\(\vec{a} \cdot \vec{a} = |\vec{a}|^2\)[/tex] (since the dot product of a vector with itself is the square of its magnitude):
[tex]\[
|\vec{a}|^2 + \vec{b} \cdot \vec{a} = 0 \tag{1}
\][/tex]
2. Step 2: Express the perpendicularity condition for [tex]\(2\vec{a} + \vec{b}\)[/tex] with [tex]\(\vec{b}\)[/tex].
If [tex]\(2\vec{a} + \vec{b}\)[/tex] is perpendicular to [tex]\(\vec{b}\)[/tex], their dot product is zero:
[tex]\[
(2\vec{a} + \vec{b}) \cdot \vec{b} = 0
\][/tex]
Expanding this:
[tex]\[
2\vec{a} \cdot \vec{b} + \vec{b} \cdot \vec{b} = 0
\][/tex]
We know that [tex]\(\vec{b} \cdot \vec{b} = |\vec{b}|^2\)[/tex]:
[tex]\[
2\vec{a} \cdot \vec{b} + |\vec{b}|^2 = 0 \tag{2}
\][/tex]
3. Step 3: Solve the system of equations derived from the dot products.
- From equation (1):
[tex]\[
|\vec{a}|^2 + \vec{a} \cdot \vec{b} = 0
\][/tex]
Solving for [tex]\(\vec{a} \cdot \vec{b}\)[/tex]:
[tex]\[
\vec{a} \cdot \vec{b} = -|\vec{a}|^2 \tag{3}
\][/tex]
- From equation (2):
[tex]\[
2(\vec{a} \cdot \vec{b}) + |\vec{b}|^2 = 0
\][/tex]
Substitute [tex]\(\vec{a} \cdot \vec{b}\)[/tex] from equation (3):
[tex]\[
2(-|\vec{a}|^2) + |\vec{b}|^2 = 0
\][/tex]
Simplify this:
[tex]\[
-2|\vec{a}|^2 + |\vec{b}|^2 = 0
\][/tex]
[tex]\[
|\vec{b}|^2 = 2|\vec{a}|^2
\][/tex]
Taking the square root of both sides:
[tex]\[
|\vec{b}| = \sqrt{2} |\vec{a}|
\][/tex]
4. Step 4: Calculate the desired ratio.
The ratio [tex]\(\frac{|\vec{b}|}{|\vec{a}|}\)[/tex] is:
[tex]\[
\frac{|\vec{b}|}{|\vec{a}|} = \frac{\sqrt{2} |\vec{a}|}{|\vec{a}|} = \sqrt{2}
\][/tex]
Therefore, the ratio [tex]\(\frac{|\vec{b}|}{|\vec{a}|}\)[/tex] is [tex]\(\boxed{\sqrt{2}}\)[/tex].
