The position vectors of P and Q are respectively `vec a and vec b `. If R is a point on `vec(PQ) ` such that `vec(PR) = 5 vec(PQ),` then the position vector of R, is

Options

A. `5 vec b - 4 vec a `
B. `5 vec b + 4 vec a `
C. `4 vec b - 5 vec a `
D. `4 vec b + 5 vec a `

Answer

To solve the problem, we need to find the position vector of point R given the position vectors of points P and Q, and the relationship between the vectors.

### Step-by-Step Solution:

1. **Identify the Position Vectors**:
Let the position vector of point P be \(\vec{a}\) and the position vector of point Q be \(\vec{b}\).

2. **Determine the Vector \(\vec{PQ}\)**:
The vector \(\vec{PQ}\) can be expressed as:
\[ \vec{PQ} = \vec{Q} - \vec{P} = \vec{b} - \vec{a} \]

3. **Express \(\vec{PR}\)**:
We know that \(\vec{PR} = 5 \vec{PQ}\). Substituting the expression for \(\vec{PQ}\):
\[ \vec{PR} = 5(\vec{b} - \vec{a}) = 5\vec{b} - 5\vec{a} \]

4. **Relate \(\vec{PR}\) to \(\vec{R}\)**:
The vector \(\vec{PR}\) can also be expressed in terms of the position vector of R:
\[ \vec{PR} = \vec{R} - \vec{P} = \vec{R} - \vec{a} \]

5. **Set the Two Expressions for \(\vec{PR}\) Equal**:
Now we can set the two expressions for \(\vec{PR}\) equal to each other:
\[ \vec{R} - \vec{a} = 5\vec{b} - 5\vec{a} \]

6. **Solve for \(\vec{R}\)**:
Rearranging the equation gives:
\[ \vec{R} = 5\vec{b} - 5\vec{a} + \vec{a} \]
Simplifying this, we have:
\[ \vec{R} = 5\vec{b} - 4\vec{a} \]

### Final Answer:
The position vector of point R is:
\[ \vec{R} = 5\vec{b} - 4\vec{a} \]

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