To solve for the value of [tex]\( |\vec{a} \cdot \vec{c}| \)[/tex], we will use the provided information about the vectors [tex]\(\vec{a}\)[/tex], [tex]\(\vec{b}\)[/tex], and [tex]\(\vec{c}\)[/tex]. Here's a step-by-step approach:
1. Given Information:
- [tex]\(\vec{a} \cdot \vec{b} = 0\)[/tex]: This means that [tex]\(\vec{a}\)[/tex] is perpendicular to [tex]\(\vec{b}\)[/tex].
- [tex]\(\vec{b} \cdot \vec{c} = 0\)[/tex]: This means that [tex]\(\vec{b}\)[/tex] is perpendicular to [tex]\(\vec{c}\)[/tex].
- The magnitude of vector [tex]\(\vec{a}\)[/tex], [tex]\( |\vec{a}| = 1 \)[/tex] unit.
- The magnitude of vector [tex]\(\vec{c}\)[/tex], [tex]\( |\vec{c}| = 6 \)[/tex] units.
2. Vectors are Coplanar:
- Since [tex]\(\vec{a}\)[/tex], [tex]\(\vec{b}\)[/tex], and [tex]\(\vec{c}\)[/tex] are coplanar and [tex]\(\vec{a}\)[/tex] and [tex]\(\vec{b}\)[/tex] are perpendicular, as well as [tex]\(\vec{b}\)[/tex] and [tex]\(\vec{c}\)[/tex] being perpendicular, it implies that [tex]\(\vec{a}\)[/tex] and [tex]\(\vec{c}\)[/tex] lie in the plane defined by [tex]\(\vec{b}\)[/tex], but they are not necessarily perpendicular to each other.
3. Dot Product:
- The dot product [tex]\(\vec{a} \cdot \vec{c} = |\vec{a}| |\vec{c}| \cos(\theta)\)[/tex], where [tex]\(\theta\)[/tex] is the angle between [tex]\(\vec{a}\)[/tex] and [tex]\(\vec{c}\)[/tex].
- We need to find the absolute value of [tex]\(\vec{a} \cdot \vec{c}\)[/tex], denoted as [tex]\( |\vec{a} \cdot \vec{c}| \)[/tex].
4. Magnitude of the Dot Product:
- Given [tex]\(|\vec{a}| = 1\)[/tex] and [tex]\(|\vec{c}| = 6\)[/tex], we can substitute these values into the formula:
[tex]\[
|\vec{a} \cdot \vec{c}| = |1 \cdot 6 \cdot \cos(\theta)|
\][/tex]
[tex]\[
|\vec{a} \cdot \vec{c}| = |6 \cos(\theta)|
\][/tex]
5. Maximum Value Consideration:
- The cosine function [tex]\(\cos(\theta)\)[/tex] ranges between [tex]\(-1\)[/tex] and [tex]\(1\)[/tex].
- Therefore, the maximum absolute value of [tex]\(6 \cos(\theta)\)[/tex] will occur when [tex]\(\cos(\theta)\)[/tex] is at its extreme values, [tex]\( \pm 1 \)[/tex].
6. Final Value:
- The maximum and minimum values of [tex]\( 6 \cos(\theta) \)[/tex] are [tex]\( \pm 6 \)[/tex].
- Therefore, the absolute maximum value is:
[tex]\[
|\vec{a} \cdot \vec{c}| = 6
\][/tex]
Thus, the value of [tex]\( |\vec{a} \cdot \vec{c}| \)[/tex] is [tex]\( 6 \)[/tex] units.
