If the points `P( veca + 2 vec b + vec c ), Q (2 veca + 3 vecb), R (vecb+ t vec c ) ` are collinear, where `veca , vec b , vec c ` are non-coplanar vectors, the value of t is

Options

A. -2
B. `-1//2`
C. `1//2`
D. 2

Explanation

To determine the value of \( t \) such that the points \( P(\vec{a} + 2\vec{b} + \vec{c}), Q(2\vec{a} + 3\vec{b}), R(\vec{b} + t\vec{c}) \) are collinear, we can use the concept that points are collinear if the vector from one point to another is a scalar multiple of the vector from a third point to one of those points.

### Step-by-Step Solution:

1. **Define the Vectors**:
- Let \( P = \vec{a} + 2\vec{b} + \vec{c} \)
- Let \( Q = 2\vec{a} + 3\vec{b} \)
- Let \( R = \vec{b} + t\vec{c} \)

2. **Find the Vectors \( \vec{PQ} \) and \( \vec{QR} \)**:
- The vector \( \vec{PQ} = Q - P \):
\[ \vec{PQ} = (2\vec{a} + 3\vec{b}) - (\vec{a} + 2\vec{b} + \vec{c}) = (2\vec{a} - \vec{a}) + (3\vec{b} - 2\vec{b}) - \vec{c} = \vec{a} + \vec{b} - \vec{c} \]
- The vector \( \vec{QR} = R - Q \):
\[ \vec{QR} = (\vec{b} + t\vec{c}) - (2\vec{a} + 3\vec{b}) = (-2\vec{a}) + (t\vec{c} - 2\vec{b}) = -2\vec{a} + (t - 2)\vec{c} \]

3. **Set Up the Collinearity Condition**:
- The points \( P, Q, R \) are collinear if \( \vec{PQ} = \lambda \vec{QR} \) for some scalar \( \lambda \).
- Thus, we have:
\[ \vec{a} + \vec{b} - \vec{c} = \lambda (-2\vec{a} + (t - 2)\vec{c}) \]

4. **Equate Coefficients**:
- From the equation above, we can equate the coefficients of \( \vec{a}, \vec{b}, \vec{c} \):
- Coefficient of \( \vec{a} \): \( 1 = -2\lambda \)
- Coefficient of \( \vec{b} \): \( 1 = 0 \) (This indicates that \( \vec{b} \) does not appear in \( \vec{QR} \), thus it must be zero)
- Coefficient of \( \vec{c} \): \( -1 = \lambda(t - 2) \)

5. **Solve for \( \lambda \)**:
- From \( 1 = -2\lambda \):
\[ \lambda = -\frac{1}{2} \]

6. **Substitute \( \lambda \) into the Coefficient of \( \vec{c} \)**:
- Substitute \( \lambda \) into \( -1 = \lambda(t - 2) \):
\[ -1 = -\frac{1}{2}(t - 2) \]
- Multiply both sides by -2:
\[ 2 = t - 2 \]
- Thus, solving for \( t \):
\[ t = 4 \]

### Final Answer:
The value of \( t \) is \( 4 \).