The scalar product of the vector $\overrightarrow a = \hat i + \hat j + \hat k$ with a unit vector along the sums of vectors $\overrightarrow b = 2 \hat i + 4\hat j - 5 \hat k$ and $\overrightarrow c = \lambda \hat i + 2 \hat j + 3 \hat k\;$ is equal to one. Find the value of $\lambda$ and hence find the unit vector along $\overrightarrow b + \overrightarrow c$

Question

$\begin{array}{1 1} \frac{3 \hat i}{7} +\frac{6 \hat j}{7} -\frac{2 \hat k}{7} \\ \frac{4 \hat i}{7} +\frac{6 \hat j}{7} -\frac{2 \hat k}{7} \\ \frac{2 \hat i}{7} +\frac{6 \hat j}{7} -\frac{ \hat k} {7} \\ \frac{3 \hat i}{7} -\frac{3 \hat j}{7} -\frac{2 \hat k}{7}\end{array}$