Express the vector ` vec a=5 hat i-2 hat j+5 hat k` as the sum of two vectors such that one is parallel to the vector ` vec b=3 hat i+ hat k` and other is perpendicular to ` vec bdot`

Explanation

To express the vector \(\vec{a} = 5 \hat{i} - 2 \hat{j} + 5 \hat{k}\) as the sum of two vectors, one parallel to the vector \(\vec{b} = 3 \hat{i} + \hat{k}\) and the other perpendicular to \(\vec{b}\), we can follow these steps:

### Step 1: Define the vectors
Let:
- \(\vec{a_1}\) be the vector parallel to \(\vec{b}\)
- \(\vec{a_2}\) be the vector perpendicular to \(\vec{b}\)

We can express \(\vec{a_1}\) as:
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