Dot products of a vector with vectors `3 hat i-5 hat k ,2 hat i+7 hat ja n d vec a= hat i+ hat j+ hat k` are respectively -1, 6 and 5. Find the vector.

Explanation

To solve the problem, we need to find the vector \(\vec{a} = x \hat{i} + y \hat{j} + z \hat{k}\) given the dot products with three other vectors. We will set up equations based on the given dot products and solve for \(x\), \(y\), and \(z\).

### Step 1: Set up the equations based on the dot products

1. The first vector is \(3 \hat{i} - 5 \hat{k}\) and the dot product with \(\vec{a}\) is given as \(-1\):
\[ \vec{a} \cdot (3 \hat{i} - 5 \hat{k}) = -1 \]
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