Let's solve this problem step-by-step given the data:
1. [tex]\(\vec{a} + \vec{b} + \vec{c} = 0\)[/tex]
2. [tex]\(|\vec{a}| = 2\)[/tex]
3. [tex]\(|\vec{b}| = 3\)[/tex]
4. [tex]\(\vec{a} \cdot \vec{b} = 6\)[/tex]
We start with equation [tex]\(\vec{a} + \vec{b} + \vec{c} = 0\)[/tex]. This can be rearranged to [tex]\(\vec{c} = -(\vec{a} + \vec{b})\)[/tex].
To find [tex]\(|\vec{c}|\)[/tex], we need to determine the magnitude of [tex]\(\vec{c}\)[/tex]. Since [tex]\(\vec{c} = -(\vec{a} + \vec{b})\)[/tex], the magnitudes are the same; the magnitude is unaffected by the negative sign:
[tex]\[ |\vec{c}| = |-(\vec{a} + \vec{b})| = |\vec{a} + \vec{b}| \][/tex]
Next, we use the properties of dot products and magnitudes to determine [tex]\(|\vec{a} + \vec{b}|\)[/tex]. The magnitude of the vector sum [tex]\(\vec{a} + \vec{b}\)[/tex] can be given by the formula:
[tex]\[ |\vec{a} + \vec{b}| = \sqrt{|\vec{a}|^2 + |\vec{b}|^2 + 2 \vec{a} \cdot \vec{b}} \][/tex]
We substitute the given values into this formula:
[tex]\[ |\vec{a} + \vec{b}| = \sqrt{(2)^2 + (3)^2 + 2 \times 6} \][/tex]
Calculate each term separately:
[tex]\[ (2)^2 = 4 \][/tex]
[tex]\[ (3)^2 = 9 \][/tex]
[tex]\[ 2 \times \vec{a} \cdot \vec{b} = 2 \times 6 = 12 \][/tex]
Thus,
[tex]\[ |\vec{a} + \vec{b}| = \sqrt{4 + 9 + 12} = \sqrt{25} = 5 \][/tex]
Since [tex]\(|\vec{c}| = |\vec{a} + \vec{b}|\)[/tex], we find:
[tex]\[ |\vec{c}| = 5 \][/tex]
Hence, the magnitude of vector [tex]\(\vec{c}\)[/tex] is:
[tex]\[ |\vec{c}| = 5 \][/tex]
