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Chapter 10 · Vector Algebra

Exercise 10.2 · Q16: Midpoint of P(2,3,4)P(2,3,4) and Q(4,1,−2)Q(4,1,-2)

Exercise 10.2Section formula: the midpoint · Section 10.5.3
The question

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

Exam answer

Here OP→=2i^+3j^+4k^\overrightarrow{OP} = 2\hat i + 3\hat j + 4\hat k and OQ→=4i^+j^−2k^\overrightarrow{OQ} = 4\hat i + \hat j - 2\hat k⁠.

OP→+OQ→=(2+4)i^+(3+1)j^+(4−2)k^=6i^+4j^+2k^\begin{aligned} \overrightarrow{OP} + \overrightarrow{OQ} &= (2 + 4)\hat i + (3 + 1)\hat j \\ &\quad + (4 - 2)\hat k \\ &= 6\hat i + 4\hat j + 2\hat k \end{aligned}

The position vector of the midpoint R of PQ is

OR→=OP→+OQ→2=6i^+4j^+2k^2\begin{aligned} \overrightarrow{OR} &= \frac{\overrightarrow{OP} + \overrightarrow{OQ}}{2} \\ &= \frac{6\hat i + 4\hat j + 2\hat k}{2} \end{aligned}

∴ OR→=3i^+2j^+k^\therefore\ \overrightarrow{OR} = 3\hat i + 2\hat j + \hat k⁠.

Want the reason behind every step, the basics and the checks?

The one idea this problem is built on

The midpoint divides PQ in the ratio 1 : 1, so the section formula gives the simple average OR→=OP→+OQ→2\overrightarrow{OR} = \frac{\overrightarrow{OP} + \overrightarrow{OQ}}{2}⁠. In components: add the matching coordinates and halve.

Given, to find, and the plan
Given
The points P(2, 3, 4) and Q(4, 1, −2-2⁠).
To find
The position vector OR→\overrightarrow{OR} of the midpoint R of PQ.
Plan
(1) Write the position vectors of P and Q. (2) Use the midpoint formula. (3) Add and halve. (4) Check that PR→=RQ→\overrightarrow{PR} = \overrightarrow{RQ}⁠.

Basics you need

1Position vector of a point

The point (x,y,z)(x, y, z) has position vector xi^+yj^+zk^x\hat i + y\hat j + z\hat k (Section 10.5.1). Basics: three numbers fix a point

2The midpoint formula

If R is the midpoint of PQ, then m=nm = n in the section formula and OR→=a⃗+b⃗2\overrightarrow{OR} = \frac{\vec a + \vec b}{2}⁠, where a⃗\vec a⁠, b⃗\vec b are the position vectors of P and Q (Remark, Section 10.5.3). Lesson: the section formula

3Working in components

Vectors are added and multiplied by numbers component by component (Section 10.5.1, (i) and (iv)). Lesson: components

Solution

  1. Write the position vectors

    OP→=2i^+3j^+4k^,OQ→=4i^+j^−2k^.\begin{aligned} \overrightarrow{OP} &= 2\hat i + 3\hat j + 4\hat k, \\ \overrightarrow{OQ} &= 4\hat i + \hat j - 2\hat k. \end{aligned}
    The position vector of a point has the coordinates of the point as its components. The question says "the vector joining the points P and Q", so the midpoint is the midpoint R of the segment PQ.
    OP→=xi^+yj^+zk^\overrightarrow{OP} = x\hat i + y\hat j + z\hat k for P(x, y, z) (Section 10.5.1, Fig 10.14). Basics: three numbers fix a point

  2. Use the midpoint formula

    OR→=OP→+OQ→2.\overrightarrow{OR} = \frac{\overrightarrow{OP} + \overrightarrow{OQ}}{2}.
    The midpoint divides PQ internally in the ratio 1 : 1. With m=n=1m = n = 1 the section formula m OQ→+n OP→m+n\frac{m\,\overrightarrow{OQ} + n\,\overrightarrow{OP}}{m + n} becomes OQ→+OP→2\frac{\overrightarrow{OQ} + \overrightarrow{OP}}{2}⁠: both ends get the same weight.
    Midpoint: Remark after Case II, Section 10.5.3. Lesson: the section formula

  3. Add and halve

    OR→=(2+4)i^+(3+1)j^+(4−2)k^2=6i^+4j^+2k^2=3i^+2j^+k^.\begin{aligned} \overrightarrow{OR} &= \frac{(2 + 4)\hat i + (3 + 1)\hat j + (4 - 2)\hat k}{2} \\ &= \frac{6\hat i + 4\hat j + 2\hat k}{2} \\ &= 3\hat i + 2\hat j + \hat k. \end{aligned}

    So OR→=3i^+2j^+k^\overrightarrow{OR} = 3\hat i + 2\hat j + \hat k, and the midpoint is R(3, 2, 1).

    Add the matching components (the z-components are 4+(−2)=24 + (-2) = 2⁠), then divide each by 2. Each coordinate of R is the average of the coordinates of P and Q.
    Sum and scalar multiple by components (Section 10.5.1, (i) and (iv)). Lesson: components

  4. Check that R is halfway

    PR→=(3−2)i^+(2−3)j^+(1−4)k^=i^−j^−3k^,RQ→=(4−3)i^+(1−2)j^+(−2−1)k^=i^−j^−3k^.\begin{aligned} \overrightarrow{PR} &= (3 - 2)\hat i + (2 - 3)\hat j + (1 - 4)\hat k \\ &= \hat i - \hat j - 3\hat k, \\ \overrightarrow{RQ} &= (4 - 3)\hat i + (1 - 2)\hat j + (-2 - 1)\hat k \\ &= \hat i - \hat j - 3\hat k. \end{aligned}

    So PR→=RQ→\overrightarrow{PR} = \overrightarrow{RQ}⁠: R lies on PQ, exactly halfway, with PR=RQ=1+1+9=11\text{PR} = \text{RQ} = \sqrt{1 + 1 + 9} = \sqrt{11}⁠.

    P(2, 3, 4) R(3, 2, 1) Q(4, 1, −2) √11 √11 |PQ| = 2√11 ≈ 6.63
    The segment PQ drawn true to scale with one scale: its length is 2112\sqrt{11}⁠, and the midpoint R(3, 2, 1) cuts it into two halves of length 11\sqrt{11}⁠.
    Equal vectors PR→\overrightarrow{PR} and RQ→\overrightarrow{RQ} mean the two pieces have the same length and the same direction, so R is on the segment and in the middle of it.
    PR→=OR→−OP→\overrightarrow{PR} = \overrightarrow{OR} - \overrightarrow{OP} (Section 10.5.2). Lesson: Examples 11–12
Result

The position vector of the midpoint of PQ is 3i^+2j^+k^3\hat i + 2\hat j + \hat k (the point R(3, 2, 1)).

(Textbook answer: 3i^+2j^+k^3\hat i + 2\hat j + \hat k⁠.)

Check it with numbers

PQ→=2i^−2j^−6k^\overrightarrow{PQ} = 2\hat i - 2\hat j - 6\hat k⁠, so ∣PQ→∣=4+4+36=44|\overrightarrow{PQ}| = \sqrt{4 + 4 + 36} = \sqrt{44} =211≈6.633250= 2\sqrt{11} \approx 6.633250⁠, and each half is 11≈3.316625\sqrt{11} \approx 3.316625⁠.

Averages of the coordinates: 2+42=3\frac{2 + 4}{2} = 3⁠, 3+12=2\frac{3 + 1}{2} = 2⁠, 4+(−2)2=1\frac{4 + (-2)}{2} = 1⁠.

Common mistakes
  1. Subtracting instead of adding. OQ→−OP→2=i^−j^−3k^\frac{\overrightarrow{OQ} - \overrightarrow{OP}}{2} = \hat i - \hat j - 3\hat k is half of PQ→\overrightarrow{PQ}⁠, the step from P to R, not the position vector of R.
  2. Forgetting to halve. 6i^+4j^+2k^6\hat i + 4\hat j + 2\hat k is OP→+OQ→\overrightarrow{OP} + \overrightarrow{OQ}⁠, twice the answer.
  3. Sign slip in the z-component. 4+(−2)=24 + (-2) = 2⁠, so the z-coordinate of R is 1, not 3.
What to take away
  1. Midpoint: OR→=OP→+OQ→2\overrightarrow{OR} = \frac{\overrightarrow{OP} + \overrightarrow{OQ}}{2}⁠, the section formula with ratio 1 : 1.
  2. In coordinates: average each coordinate.
  3. Check with PR→=RQ→\overrightarrow{PR} = \overrightarrow{RQ}⁠.

Every fact used, and where it comes from

FactUsed inSource
Position vector xi^+yj^+zk^x\hat i + y\hat j + z\hat k of P(x, y, z)Step 1Section 10.5.1, Fig 10.14 (Basics: three numbers fix a point)
Midpoint: OR→=a⃗+b⃗2\overrightarrow{OR} = \frac{\vec a + \vec b}{2}Step 2Remark, Section 10.5.3 (Lesson: the section formula)
Sum and scalar multiple by componentsStep 3Section 10.5.1, (i) and (iv)
Vector joining two points; magnitudeStep 4Sections 10.5.2 and 10.5.1
The answerResultNCERT answer key, Exercise 10.2 Q16 (Lesson: components, lengths and unit vectors)
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