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 find the position vector of point R, we will follow these steps:

### Step 1: Understand the given information
Let the position vectors of points P and Q be represented as:
- \( \vec{P} = \vec{a} \)
- \( \vec{Q} = \vec{b} \)

The vector \( \vec{PQ} \) can be expressed as:
\[ \vec{PQ} = \vec{Q} - \vec{P} = \vec{b} - \vec{a} \]

### Step 2: Express \( \vec{PR} \) in terms of \( \vec{PQ} \)
We know that \( \vec{PR} = 5 \vec{PQ} \). Therefore, we can write:
\[ \vec{PR} = 5(\vec{Q} - \vec{P}) = 5(\vec{b} - \vec{a}) \]

### Step 3: Relate \( \vec{R} \) to \( \vec{P} \) and \( \vec{Q} \)
Using the definition of \( \vec{PR} \):
\[ \vec{R} - \vec{P} = \vec{PR} \]
Substituting the expression for \( \vec{PR} \):
\[ \vec{R} - \vec{a} = 5(\vec{b} - \vec{a}) \]

### Step 4: Solve for \( \vec{R} \)
Rearranging the equation gives:
\[ \vec{R} = \vec{a} + 5(\vec{b} - \vec{a}) \]
Expanding this:
\[ \vec{R} = \vec{a} + 5\vec{b} - 5\vec{a} = 5\vec{b} - 4\vec{a} \]

### Step 5: Final expression for \( \vec{R} \)
Thus, the position vector of point R is:
\[ \vec{R} = 5\vec{b} - 4\vec{a} \]

### Summary
The position vector of point R is \( \vec{R} = 5\vec{b} - 4\vec{a} \).

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