Vector projection and cross product (Jee Mains)

Scalar and Vector Products

Medium

Vector projection and cross product

Let $\vec a = \hat i - 2\hat j + 3\hat k,$ $\vec b = 2\hat i + \hat j - \hat k,$ $\vec c = \lambda \hat i + \hat j + \hat k$ and $\vec \nu = \vec a \times \vec b$ . If $\vec \nu .\vec c = 11$ and the length of the projection of $\vec b$ on $\vec c$ is p, then 9p² is equal to :

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