Find "a" so that the vectors `vecp = 3 hati - 2hatj and vecq = 2 hati + a hatj` be orthogonal.

Answer

To find the value of \( a \) such that the vectors \( \vec{p} = 3 \hat{i} - 2 \hat{j} \) and \( \vec{q} = 2 \hat{i} + a \hat{j} \) are orthogonal, we can follow these steps:

### Step 1: Understand the condition for orthogonality
Two vectors are orthogonal if their dot product is zero. Therefore, we need to set up the equation:
\[ \vec{p} \cdot \vec{q} = 0 \]

### Step 2: Write the dot product of the vectors
Substituting the given vectors into the dot product formula:
\[ (3 \hat{i} - 2 \hat{j}) \cdot (2 \hat{i} + a \hat{j}) = 0 \]

### Step 3: Calculate the dot product
Using the distributive property of the dot product:
\[ = 3 \cdot 2 + (-2) \cdot a = 6 - 2a \]

### Step 4: Set the dot product equal to zero
Now, we set the result of the dot product equal to zero:
\[ 6 - 2a = 0 \]

### Step 5: Solve for \( a \)
Rearranging the equation gives:
\[ 6 = 2a \]
Dividing both sides by 2:
\[ a = 3 \]

### Final Answer
Thus, the value of \( a \) is \( 3 \).

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