Vectors · Cross product and dot product

Let

\vec a = 2\hat i - \hat j + \hat k

and

\vec b = \lambda\hat j + 2\hat k

\lambda \in \mathbb{Z}

be two vectors, Let

\vec c = \vec a\times\vec b

and

\vec d

be a vector of magnitude 2 in

yz

-plane. If

\lvert\vec c\rvert = \sqrt{53}

, then the maximum possible value of

\left(\vec c\cdot\vec d\right)^2

is equal to:

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