Question 19 of 68beginner🔧 ApplyNumerical2 marks

Find the position vector of the mid point of the vector joining the points P(2, 3, 4) and Q(4, 1, -2).

Correct Answer

The position vector of the midpoint is 3i^+2j^+k^3\hat{i} + 2\hat{j} + \hat{k}.

Exercise: EXERCISE 10.2 | Q: 16 | (Page 18)
For More Understanding

Explanation

This question tests the application of the midpoint formula for position vectors. According to the Remark in the Section formula (10.5.3), if R is the midpoint of PQ, then the position vector is given by OR→=a⃗+b⃗2\overrightarrow{OR} = \frac{\vec{a} + \vec{b}}{2}, where a⃗\vec{a} and b⃗\vec{b} are the position vectors of P and Q respectively. The student needs to first write the position vectors of both points and then apply the midpoint formula.

Solution Steps

  1. Step 1: Write the position vectors of points P and Q. Position vector of P(2, 3, 4) is OP⃗=2i^+3j^+4k^\vec{OP} = 2\hat{i} + 3\hat{j} + 4\hat{k} Position vector of Q(4, 1, -2) is OQ⃗=4i^+j^−2k^\vec{OQ} = 4\hat{i} + \hat{j} - 2\hat{k}

  2. Step 2: Apply the midpoint formula. If R is the midpoint of PQ, then the position vector of R is:

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

  3. Step 3: Calculate the sum of position vectors.

    OP⃗+OQ⃗=(2i^+3j^+4k^)+(4i^+j^−2k^)\vec{OP} + \vec{OQ} = (2\hat{i} + 3\hat{j} + 4\hat{k}) + (4\hat{i} + \hat{j} - 2\hat{k}) =(2+4)i^+(3+1)j^+(4−2)k^= (2+4)\hat{i} + (3+1)\hat{j} + (4-2)\hat{k} =6i^+4j^+2k^= 6\hat{i} + 4\hat{j} + 2\hat{k}

  4. Step 4: Divide by 2 to get the midpoint position vector.

    OR→=6i^+4j^+2k^2=3i^+2j^+k^\overrightarrow{OR} = \frac{6\hat{i} + 4\hat{j} + 2\hat{k}}{2} = 3\hat{i} + 2\hat{j} + \hat{k}
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