\(\vec a = 2\hat i + \hat j - \hat k\)
\(\vec b = \hat i + 2\hat j - \hat k\)
\(\vec c = \alpha \,\vec a + \beta\,\vec b, \alpha, \beta \in\,R\)
Projection of \(\vec c\)on the vector \((\vec a + \vec b)\)
\(= \frac{\vec c . (\vec a + \vec b)}{|\vec a + \vec b|}\)
\(= \frac{(\alpha \,\vec a+ \beta\, \vec b).(\vec a + \vec b)}{|\vec a + \vec b|}\)
\(= \frac{\alpha (\vec a)^2 + (\alpha + \beta ) \,\vec a . \vec b + \beta |\vec b|^2}{|\vec a + \vec b |}\)
\(= \frac{6 \alpha + 3(\alpha + \beta ) + 6\beta}{|3\hat i + 3\hat j|}\) \(\begin{pmatrix}\therefore |\vec a|^2 = 2^2 + 1^2 + (-1)^2 = 6\\
|\vec b|^2 = 1^2 + 2^2 + 1^2 = 6\\\vec a .\vec b = (2\hat i+ \hat j - \hat k).(\hat i + 2\hat j+ \hat k)\\= 2+2-1 =3\end {pmatrix}\)
\(= \frac{9(\alpha + \beta)}{\sqrt{9+9}}\)
\(=\frac{3}{\sqrt{2}}(\alpha + \beta)\)
Given that projection of \(\vec c\)on \((\vec a + \vec b)\) is 3√2.
\(\therefore \frac{3}{\sqrt2}(\alpha + \beta) = 3\sqrt2\)
⇒ \(\alpha + \beta = 2 \) .......(i)
Now,
\([\vec c - (\vec a \times \vec b )]. \vec c = \vec c . \vec c - (\vec a \times \vec b). \vec c \)
\( = |\vec c|^2 - [\vec c \,\,\vec a\,\,\vec b] \) \((\because \vec a.(\vec b \times \vec c) = [\vec a \,\, \vec b\,\, \vec c])\)
\(= |\alpha \vec a + \beta \vec b|^2 - [\alpha \vec a \,+\beta \vec b \,\,\,\,\vec a \,\,\,\, \vec b]\)
\( = \alpha^2 |\vec a|^2 + \beta^2 |\vec b|^2 + 2 \alpha \beta \, \vec a. \vec b - \alpha[\vec a \,\,\,\, \vec a \,\,\,\,\vec b] - \beta[\vec b \,\,\,\,\vec a\,\,\,\,\vec b]\)
\(= 6\alpha^2+ 6\beta ^2 + 6\alpha \beta - 0\) \((\because [\vec a \,\,\,\,\vec a\,\,\,\,\vec b]= 0) \)
\( = 6 (\alpha ^2 + \beta ^2 + \alpha\beta)\)
\(= 6[(\alpha + \beta)^2 - \alpha \beta]\)
\( = 6 (\alpha + \beta )^2 - 6 \alpha\beta\)
\(= 6 (2)^2 - 6\alpha \beta\) \((\because \alpha + \beta = 2 [\text{from (i)}])\)
\( = 24 - 6\alpha \beta\)
\(\because \alpha + \beta = 2\)
⇒ \((\alpha + \beta)^2 = 4\)
⇒ \(\alpha ^2 + \beta^2 + 2\alpha\beta = 4\)
⇒ \(2 \alpha \beta = 4 -(\alpha^2 + \beta^2)\)
⇒ \(2\alpha \beta \le 4 \) \((\because \alpha^2 + \beta^2\ge0)\)
⇒ \(\alpha \beta \le 2 \)
⇒ \(-\alpha \beta \ge -2 \)
\(\therefore [\vec c - (\vec a \times \vec b)].\vec c = 24 - 6 \alpha \beta\)
\(\ge 24 - 12\)
⇒ \([\vec c - (\vec a \times \vec b)].\vec c \ge 12\)
\(\therefore \) Minimum value is 12.