If `vec(A)` and `vec(B)` are perpendicular Vectors and vector `vec(A)= 5hat(i)+7hat(j)-3hat(k)` and `vec(B)= 2hat(i)+2hat(j)-ahat(k)`. The value of `a` is

Options

A. `-2`
B. 8
C. `-7`
D. `-8`

Answer

To find the value of \( a \) such that the vectors \( \vec{A} \) and \( \vec{B} \) are perpendicular, we will follow these steps:

### Step 1: Write down the vectors
We have:
\[ \vec{A} = 5\hat{i} + 7\hat{j} - 3\hat{k} \]
\[ \vec{B} = 2\hat{i} + 2\hat{j} - a\hat{k} \]

### Step 2: Use the condition for perpendicular vectors
Two vectors are perpendicular if their dot product is zero:
\[ \vec{A} \cdot \vec{B} = 0 \]

### Step 3: Calculate the dot product
The dot product \( \vec{A} \cdot \vec{B} \) is calculated as follows:
\[ \vec{A} \cdot \vec{B} = (5\hat{i} + 7\hat{j} - 3\hat{k}) \cdot (2\hat{i} + 2\hat{j} - a\hat{k}) \]
Using the distributive property of the dot product:
\[ = 5 \cdot 2 + 7 \cdot 2 + (-3) \cdot (-a) \]
\[ = 10 + 14 + 3a \]
\[ = 24 + 3a \]

### Step 4: Set the dot product equal to zero
Since the vectors are perpendicular:
\[ 24 + 3a = 0 \]

### Step 5: Solve for \( a \)
Rearranging the equation gives:
\[ 3a = -24 \]
\[ a = \frac{-24}{3} \]
\[ a = -8 \]

### Conclusion
The value of \( a \) is \( -8 \).

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