Question 58 of 68intermediate🔧 ApplyLong Answer4 marks

Find the position vector of a point RR which divides the line joining two points PP and QQ whose position vectors are (2a⃗+b⃗)(2\vec{a} + \vec{b}) and (a⃗−3b⃗)(\vec{a} - 3\vec{b}) externally in the ratio 1:21:2. Also, show that PP is the mid point of the line segment RQRQ.

Correct Answer

The position vector of point RR is 3a⃗+5b⃗3\vec{a} + 5\vec{b}.

Proof that P is the midpoint of RQ:

For PP to be the midpoint of RQRQ, we must have:

OP⃗=OR⃗+OQ⃗2\vec{OP} = \frac{\vec{OR} + \vec{OQ}}{2}

Substitution:

OR⃗+OQ⃗2=(3a⃗+5b⃗)+(a⃗−3b⃗)2\frac{\vec{OR} + \vec{OQ}}{2} = \frac{(3\vec{a} + 5\vec{b}) + (\vec{a} - 3\vec{b})}{2} =4a⃗+2b⃗2=2a⃗+b⃗=OP⃗= \frac{4\vec{a} + 2\vec{b}}{2} = 2\vec{a} + \vec{b} = \vec{OP}

Hence, PP is the midpoint of the line segment RQRQ.

Exercise: MISCELLANEOUS EXERCISE ON CHAPTER 10 | Q: 9 | (Page 35)
For More Understanding

Explanation

This question tests the application of the external section formula for vectors. The textbook provides the formula for external division: when RR divides PQPQ externally in the ratio m:nm:n, the position vector is (mb⃗−na⃗)/(m−n)(m\vec{b} - n\vec{a})/(m-n).

After finding RR's position vector, we verify PP is the midpoint by checking if the midpoint formula (R+Q)/2(R+Q)/2 equals PP's position vector.

Solution Steps

  1. Step 1: Identify given values — Position vector of P=2a⃗+b⃗P = 2\vec{a} + \vec{b}, Position vector of Q=a⃗−3b⃗Q = \vec{a} - 3\vec{b}, Ratio m:n=1:2m:n = 1:2 (external division)

  2. Step 2: Apply external division formula:

    OR⃗=m⋅OQ⃗−n⋅OP⃗m−n\vec{OR} = \frac{m \cdot \vec{OQ} - n \cdot \vec{OP}}{m - n}

  3. Step 3: Substitute values:

    OR⃗=1(a⃗−3b⃗)−2(2a⃗+b⃗)1−2\vec{OR} = \frac{1(\vec{a} - 3\vec{b}) - 2(2\vec{a} + \vec{b})}{1 - 2}

  4. Step 4: Simplify:

    OR⃗=a⃗−3b⃗−4a⃗−2b⃗−1=−3a⃗−5b⃗−1=3a⃗+5b⃗\vec{OR} = \frac{\vec{a} - 3\vec{b} - 4\vec{a} - 2\vec{b}}{-1} = \frac{-3\vec{a} - 5\vec{b}}{-1} = 3\vec{a} + 5\vec{b}

  5. Step 5: To prove PP is midpoint of RQRQ, verify:

    OP⃗=OR⃗+OQ⃗2\vec{OP} = \frac{\vec{OR} + \vec{OQ}}{2}

  6. Step 6: Calculate:

    OR⃗+OQ⃗2=(3a⃗+5b⃗)+(a⃗−3b⃗)2=4a⃗+2b⃗2=2a⃗+b⃗=OP⃗\frac{\vec{OR} + \vec{OQ}}{2} = \frac{(3\vec{a} + 5\vec{b}) + (\vec{a} - 3\vec{b})}{2} = \frac{4\vec{a} + 2\vec{b}}{2} = 2\vec{a} + \vec{b} = \vec{OP}

    ∴ Verified

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