The component of vector `vec(A) = a_(x) hat(i) + a_(y) hat(j) + a_(z) hat(k)` along the direction of `hat(j) - hat(k)` is

Options

A. `a_(x) - a_(y) +a_(z)`
B. `a_(z) - a_(y)`
C. `(a_(x) - a_(y)) //sqrt2`
D. `(a_(y) - a_(z))/(sqrt2)`

Answer

To find the component of the vector \( \vec{A} = a_x \hat{i} + a_y \hat{j} + a_z \hat{k} \) along the direction of \( \hat{j} - \hat{k} \), we can follow these steps:

### Step 1: Identify the Direction Vector
The direction we need to consider is given by \( \hat{j} - \hat{k} \). We can denote this direction vector as \( \vec{B} \):
\[ \vec{B} = \hat{j} - \hat{k} \]

### Step 2: Calculate the Magnitude of the Direction Vector
To find the unit vector in the direction of \( \vec{B} \), we first calculate its magnitude:
\[ |\vec{B}| = \sqrt{(0)^2 + (1)^2 + (-1)^2} = \sqrt{0 + 1 + 1} = \sqrt{2} \]

### Step 3: Find the Unit Vector in the Direction of \( \vec{B} \)
Now, we can find the unit vector \( \hat{b} \):
\[ \hat{b} = \frac{\vec{B}}{|\vec{B}|} = \frac{\hat{j} - \hat{k}}{\sqrt{2}} = \frac{1}{\sqrt{2}} \hat{j} - \frac{1}{\sqrt{2}} \hat{k} \]

### Step 4: Calculate the Dot Product of \( \vec{A} \) and \( \hat{b} \)
The component of \( \vec{A} \) along the direction of \( \hat{b} \) can be found using the dot product:
\[ \text{Component of } \vec{A} \text{ along } \hat{b} = \vec{A} \cdot \hat{b} \]
Substituting \( \vec{A} \) and \( \hat{b} \):
\[ \vec{A} \cdot \hat{b} = (a_x \hat{i} + a_y \hat{j} + a_z \hat{k}) \cdot \left(\frac{1}{\sqrt{2}} \hat{j} - \frac{1}{\sqrt{2}} \hat{k}\right) \]

### Step 5: Perform the Dot Product
Calculating the dot product:
\[ \vec{A} \cdot \hat{b} = a_x \hat{i} \cdot \left(\frac{1}{\sqrt{2}} \hat{j} - \frac{1}{\sqrt{2}} \hat{k}\right) + a_y \hat{j} \cdot \left(\frac{1}{\sqrt{2}} \hat{j} - \frac{1}{\sqrt{2}} \hat{k}\right) + a_z \hat{k} \cdot \left(\frac{1}{\sqrt{2}} \hat{j} - \frac{1}{\sqrt{2}} \hat{k}\right) \]
Since \( \hat{i} \cdot \hat{j} = 0 \) and \( \hat{i} \cdot \hat{k} = 0 \):
\[ = 0 + a_y \cdot \frac{1}{\sqrt{2}} - a_z \cdot \frac{1}{\sqrt{2}} = \frac{a_y - a_z}{\sqrt{2}} \]

### Step 6: Final Result
Thus, the component of the vector \( \vec{A} \) along the direction of \( \hat{j} - \hat{k} \) is:
\[ \frac{a_y - a_z}{\sqrt{2}} \]