For component of vector `vec(A)=(3hat(i)+4hat(j)-5hat(k))`, match the following table:
`{:(,"Column I",,,"Column II"),((A),"Along y-axis",,(p),5 "unit"),((B),"Along another vector"(2hat(i)+hat(j)+2hat(k)),,(q),4 "unit"),((C),"Along another vector" (6hat(i)+8hat(j)-10hat(k)),,(r),"Zero"),((D),"Along another vector" (-3hat(i)-4hat(j)+5hat(k)),,(s),"None"):}`

Answer

To solve the problem of matching the components of the vector \(\vec{A} = 3\hat{i} + 4\hat{j} - 5\hat{k}\) with the given options in the table, we will compute the component of \(\vec{A}\) along various vectors provided in Column II. The formula to find the component of vector \(\vec{A}\) along a vector \(\vec{r}\) is given by:

\[ \text{Component of } \vec{A} \text{ along } \vec{r} = \frac{\vec{A} \cdot \vec{r}}{|\vec{r}|} \]

where \(\vec{A} \cdot \vec{r}\) is the dot product of the vectors and \(|\vec{r}|\) is the magnitude of vector \(\vec{r}\).

### Step 1: Component along y-axis
1. **Identify \(\vec{r}\)**: For the y-axis, \(\vec{r} = \hat{j}\).
2. **Calculate the dot product**:
\[ \vec{A} \cdot \vec{r} = (3\hat{i} + 4\hat{j} - 5\hat{k}) \cdot \hat{j} = 4 \]
3. **Calculate the magnitude of \(\vec{r}\)**:
\[ |\vec{r}| = |\hat{j}| = 1 \]
4. **Calculate the component**:
\[ \text{Component} = \frac{4}{1} = 4 \text{ units} \]
**Match with Column II**: This corresponds to option \(q\).

### Step 2: Component along vector \(2\hat{i} + \hat{j} + 2\hat{k}\)
1. **Identify \(\vec{r}\)**: \(\vec{r} = 2\hat{i} + \hat{j} + 2\hat{k}\).
2. **Calculate the dot product**:
\[ \vec{A} \cdot \vec{r} = (3\hat{i} + 4\hat{j} - 5\hat{k}) \cdot (2\hat{i} + \hat{j} + 2\hat{k}) = 3 \cdot 2 + 4 \cdot 1 - 5 \cdot 2 = 6 + 4 - 10 = 0 \]
3. **Calculate the magnitude of \(\vec{r}\)**:
\[ |\vec{r}| = \sqrt{2^2 + 1^2 + 2^2} = \sqrt{4 + 1 + 4} = \sqrt{9} = 3 \]
4. **Calculate the component**:
\[ \text{Component} = \frac{0}{3} = 0 \]
**Match with Column II**: This corresponds to option \(r\).

### Step 3: Component along vector \(6\hat{i} + 8\hat{j} - 10\hat{k}\)
1. **Identify \(\vec{r}\)**: \(\vec{r} = 6\hat{i} + 8\hat{j} - 10\hat{k}\).
2. **Calculate the dot product**:
\[ \vec{A} \cdot \vec{r} = (3\hat{i} + 4\hat{j} - 5\hat{k}) \cdot (6\hat{i} + 8\hat{j} - 10\hat{k}) = 3 \cdot 6 + 4 \cdot 8 - 5 \cdot (-10) = 18 + 32 + 50 = 100 \]
3. **Calculate the magnitude of \(\vec{r}\)**:
\[ |\vec{r}| = \sqrt{6^2 + 8^2 + (-10)^2} = \sqrt{36 + 64 + 100} = \sqrt{200} = 10\sqrt{2} \]
4. **Calculate the component**:
\[ \text{Component} = \frac{100}{10\sqrt{2}} = \frac{10}{\sqrt{2}} = 5\sqrt{2} \]
**Match with Column II**: This corresponds to option \(p\).

### Step 4: Component along vector \(-3\hat{i} - 4\hat{j} + 5\hat{k}\)
1. **Identify \(\vec{r}\)**: \(\vec{r} = -3\hat{i} - 4\hat{j} + 5\hat{k}\).
2. **Calculate the dot product**:
\[ \vec{A} \cdot \vec{r} = (3\hat{i} + 4\hat{j} - 5\hat{k}) \cdot (-3\hat{i} - 4\hat{j} + 5\hat{k}) = 3 \cdot (-3) + 4 \cdot (-4) + (-5) \cdot 5 = -9 - 16 - 25 = -50 \]
3. **Calculate the magnitude of \(\vec{r}\)**:
\[ |\vec{r}| = \sqrt{(-3)^2 + (-4)^2 + 5^2} = \sqrt{9 + 16 + 25} = \sqrt{50} \]
4. **Calculate the component**:
\[ \text{Component} = \frac{-50}{\sqrt{50}} = -\sqrt{50} \]
**Match with Column II**: This corresponds to option \(s\).

### Final Matching:
- \(A\) (Along y-axis) matches with \(q\) (4 units)
- \(B\) (Along \(2\hat{i} + \hat{j} + 2\hat{k}\)) matches with \(r\) (0)
- \(C\) (Along \(6\hat{i} + 8\hat{j} - 10\hat{k}\)) matches with \(p\) (5 units)
- \(D\) (Along \(-3\hat{i} - 4\hat{j} + 5\hat{k}\)) matches with \(s\) (None)