Answer
$185$
Work Step by Step
Recall the dot product property for n-dimensional vectors, which states that if $\vec{u}=(u_1,u_2.....u_n)$ and $\vec{v}=(v_1,v_2.....v_n)$, then their dot product is:
$\vec{u}\cdot \vec{v}=u_1v_1+u_2v_2+........u_nv_n$
We are given that $\vec{a}=2j+k = \lt 0,2,1\gt$ and $\vec{c}=i+6j =\lt 1,6,0\gt$
Our aim is to find the dot product $[(\vec{c} \cdot \vec{c}) \vec{a}]\cdot \vec{a}$.
Now, $\vec{c}\cdot \vec{c}=\lt 1,6,0\gt \cdot \lt 1,6,0\gt=(1)(1)+(6)(6)+(0)(0) =37$
Therefore, we have:
$[(\vec{c} \cdot \vec{c}) \vec{a}]\cdot \vec{a}= [ 37 \lt 0,2,1 \gt] \cdot \lt0,2,1\gt\\=\lt0,74, 37 \gt \cdot \lt0,2,1\gt \\=(0)(0)+(74)(2)+(37)(1) \\=185$
