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Let $\vec a = \hat i + \hat j + \hat k$ , $\vec b = 2\hat i + 2\hat j + \hat k$ and $\vec d = \vec a \times \vec b$ . If $\vec c$ is a vector such that $\vec a.\vec c = \left| {\vec c} \right|,$ ${\left| {\vec c - 2\vec a} \right|^2}$ = 8 and the angle between $\vec d$ and $\vec c$ is $\frac{\pi }{4}$ , then $\left| {10 - 3\vec b.\vec c} \right| + {\left| {\vec d \times \vec c} \right|^2}$ is equal to …..

Vector Dot and Cross Product Relations