Consider the following for the next two items that follows
Let `vec(a) = hat(i) + hat(j), vec(b)= 3hat(i) + 4hat(k) and vec(b) =vec(c ) +vec(d)`, where `vec(c )` is parallel to `vec(a) and vec(d)` is perpendicular to `vec(a)`
What is `vec(c )` equal to ?

Options

A. `(3 (hat(i) + hat(j)))/(2)`
B. `(2 (hat(i) + hat(j)))/(3)`
C. `((hat(i) + hat(j)))/(2)`
D. `((hat(i) + hat(j)))/(3)`

Answer

To find the vector \(\vec{c}\), we can follow these steps:

### Step 1: Understand the given vectors
We have:
\[ \vec{a} = \hat{i} + \hat{j} \]
\[ \vec{b} = 3\hat{i} + 4\hat{k} \]
And we know:
\[ \vec{b} = \vec{c} + \vec{d} \]
where \(\vec{c}\) is parallel to \(\vec{a}\) and \(\vec{d}\) is perpendicular to \(\vec{a}\).

### Step 2: Calculate the dot product
Taking the dot product of both sides of the equation \(\vec{b} = \vec{c} + \vec{d}\) with \(\vec{a}\):
\[ \vec{a} \cdot \vec{b} = \vec{a} \cdot \vec{c} + \vec{a} \cdot \vec{d} \]

### Step 3: Compute \(\vec{a} \cdot \vec{b}\)
Calculating \(\vec{a} \cdot \vec{b}\):
\[ \vec{a} \cdot \vec{b} = (\hat{i} + \hat{j}) \cdot (3\hat{i} + 4\hat{k}) = 3(\hat{i} \cdot \hat{i}) + 4(\hat{j} \cdot \hat{k}) = 3 \cdot 1 + 0 = 3 \]

### Step 4: Substitute into the dot product equation
Now substituting back into the equation:
\[ 3 = \vec{a} \cdot \vec{c} + \vec{a} \cdot \vec{d} \]

### Step 5: Analyze the components
Since \(\vec{c}\) is parallel to \(\vec{a}\), we can express \(\vec{c}\) as:
\[ \vec{c} = k(\hat{i} + \hat{j}) \]
for some scalar \(k\).

### Step 6: Calculate \(\vec{a} \cdot \vec{c}\)
Now, calculating \(\vec{a} \cdot \vec{c}\):
\[ \vec{a} \cdot \vec{c} = (\hat{i} + \hat{j}) \cdot (k(\hat{i} + \hat{j})) = k(\hat{i} \cdot \hat{i} + \hat{j} \cdot \hat{j}) = k(1 + 1) = 2k \]

### Step 7: Substitute back into the equation
Now substituting into the equation:
\[ 3 = 2k + \vec{a} \cdot \vec{d} \]
Since \(\vec{d}\) is perpendicular to \(\vec{a}\), \(\vec{a} \cdot \vec{d} = 0\). Therefore:
\[ 3 = 2k \implies k = \frac{3}{2} \]

### Step 8: Find \(\vec{c}\)
Now substituting \(k\) back into the expression for \(\vec{c}\):
\[ \vec{c} = \frac{3}{2}(\hat{i} + \hat{j}) = \frac{3}{2}\hat{i} + \frac{3}{2}\hat{j} \]

### Final Answer
Thus, the vector \(\vec{c}\) is:
\[ \vec{c} = \frac{3}{2} \hat{i} + \frac{3}{2} \hat{j} \]

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