A straight line is given by `r=(1+t)i+3tj+(1-t)k`, where `tinR`. If this line lies in th plane `x+y+cz=d`, then the value of `(c+d)` is

Options

A. (a) `-1`
B. (b) `1`
C. (c) `7`
D. (d) `9`

Answer

To solve the problem, we need to determine the values of \(c\) and \(d\) such that the line given by the equation \( \mathbf{r} = (1+t)\mathbf{i} + 3t\mathbf{j} + (1-t)\mathbf{k} \) lies in the plane defined by the equation \( x + y + cz = d \).

### Step-by-Step Solution:

1. **Identify the line's vector form**:
The line is given as:
\[ \mathbf{r} = (1+t)\mathbf{i} + 3t\mathbf{j} + (1-t)\mathbf{k} \]
This can be rewritten in the standard form:
\[ \mathbf{r} = \mathbf{a} + t\mathbf{b} \]
where \( \mathbf{a} = \mathbf{i} + 0\mathbf{j} + \mathbf{k} \) and \( \mathbf{b} = \mathbf{i} + 3\mathbf{j} - \mathbf{k} \).

2. **Extract the parameters**:
From the line, we have:
- \( \mathbf{a} = (1, 0, 1) \)
- \( \mathbf{b} = (1, 3, -1) \)

3. **Identify the plane's normal vector**:
The plane is given by the equation:
\[ x + y + cz = d \]
The normal vector \( \mathbf{n} \) to the plane can be expressed as:
\[ \mathbf{n} = (1, 1, c) \]

4. **Condition for the line to lie in the plane**:
For the line to lie in the plane, two conditions must be satisfied:
- The dot product of \( \mathbf{b} \) and \( \mathbf{n} \) must be zero:
\[ \mathbf{b} \cdot \mathbf{n} = 0 \]
- The dot product of \( \mathbf{a} \) and \( \mathbf{n} \) must equal \( d \):
\[ \mathbf{a} \cdot \mathbf{n} = d \]

5. **Calculate the first condition**:
Compute \( \mathbf{b} \cdot \mathbf{n} \):
\[ (1, 3, -1) \cdot (1, 1, c) = 1 \cdot 1 + 3 \cdot 1 - 1 \cdot c = 1 + 3 - c = 4 - c \]
Set this equal to zero:
\[ 4 - c = 0 \implies c = 4 \]

6. **Calculate the second condition**:
Compute \( \mathbf{a} \cdot \mathbf{n} \):
\[ (1, 0, 1) \cdot (1, 1, c) = 1 \cdot 1 + 0 \cdot 1 + 1 \cdot c = 1 + 0 + c = 1 + c \]
Set this equal to \( d \):
\[ d = 1 + c = 1 + 4 = 5 \]

7. **Find \( c + d \)**:
Now that we have \( c = 4 \) and \( d = 5 \):
\[ c + d = 4 + 5 = 9 \]

### Final Answer:
The value of \( c + d \) is \( \boxed{9} \).