The vector ` vec b=3 hat i+4 hat k` is to be written as the sum of a vector ` vecalpha` parallel to ` vec a= hat i+ hat j` and a vector ` vecbeta` perpendicular to ` vec adot` Then ` vecalpha=` `3/2( hat i+ hat j)` b. `2/3( hat i+ hat j)` c. `1/2( hat i+ hat j)` d. `1/3( hat i+ hat j)`

Explanation

To solve the problem, we need to express the vector \( \vec{b} = 3 \hat{i} + 4 \hat{k} \) as the sum of two vectors: \( \vec{\alpha} \) which is parallel to \( \vec{a} = \hat{i} + \hat{j} \) and \( \vec{\beta} \) which is perpendicular to \( \vec{a} \).

### Step-by-step Solution:

1. **Define the vectors**:
- Given \( \vec{b} = 3 \hat{i} + 4 \hat{k} \)
- Given \( \vec{a} = \hat{i} + \hat{j} \)

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