Q.Find the position vector of mid-point M joining the points L(7,−6,12) and N(5,4,−2).
Concept understanding — Section Formula for Vectors
If A(a) and B(b) are two points and R(r) divides the segment AB internally in the ratio m:n (so AR:RB = m:n), then r = (m b + n a)/(m+n); if R divides AB externally in the ratio m:n, then r = (m b - n a)/(m-n). Taking m=n=1 in the internal formula gives the midpoint formula, r = (a+b)/2. Applying the section formula twice along the medians of a triangle ABC shows they all meet at the centroid G, whose position vector is g = (a+b+c)/3, dividing each median internally in the ratio 2:1; the same idea extended to a tetrahedron ABCD gives its centroid as g = (a+b+c+d)/4. A similar weighted-average construction, using the side lengths as weights, locates the incenter of a triangle. The section formula is the single tool behind almost every "find the point that divides..." or "find the centroid/fourth vertex..." question in the chapter.
The midpoint is the average of the two position vectors.
M≡(6,−1,5)
For L(7,−6,12) and N(5,4,−2), the midpoint formula gives
M=2L+N=(27+5,2−6+4,212−2)=(6,−1,5).
M≡(6,−1,5)
Arithmetic slip when averaging negative coordinates such as (-6+4)/2.
- CBSE 2024Set ANNUAL2 marksQ.If aˉ,bˉ,cˉ are the position vectors of the points A,B,C respectively and 5aˉ−3bˉ−2cˉ=0ˉ, then find the ratio in which the point C divides the line segment BA.
›Reveal solutionSolution
Rewrite 5aˉ−3bˉ−2cˉ=0 as cˉ=5−35aˉ−3bˉ, the external-division form.
5aˉ−3bˉ−2cˉ=0ˉ⇒cˉ=25aˉ−3bˉ
The external division of segment BA in ratio m:n gives point m−nmaˉ−nbˉ. Comparing 5−35aˉ−3bˉ=25aˉ−3bˉ, we get m=5,n=3.
So C divides BA externally in the ratio 5:3.
✓Final answerC divides BA externally in the ratio 5:3.
- CBSE 2023Set ANNUAL2 marksQ.If aˉ,bˉ,cˉ are the position vectors of the points A, B, C respectively and 5aˉ+3bˉ−8cˉ=0ˉ then find the ratio in which the point C divides the line segment AB.
›Reveal solutionSolution
Rewrite 5aˉ+3bˉ−8cˉ=0 as cˉ=5+35aˉ+3bˉ and read off the section-formula weights.
5aˉ+3bˉ−8cˉ=0ˉ⇒cˉ=85aˉ+3bˉ
By the section formula, a point dividing AB in ratio m:n (from A) has position vector m+nnaˉ+mbˉ. Comparing, n=5, m=3.
So C divides AB internally in the ratio 3:5.
✓Final answerC divides AB internally in the ratio 3:5.
- CBSE 2016Set ANNUAL2 marksQ.If pˉ=i^−2j^+k^ and qˉ=i^+4j^−2k^ are position vectors (P.V.) of points P and Q, find the position vector of the point R which divides segment PQ internally in the ratio 2:1.
›Reveal solutionSolution
Use the internal section formula R=m+nmqˉ+npˉ for ratio m:n=2:1.
Given pˉ=i^−2j^+k^ (P.V. of P) and qˉ=i^+4j^−2k^ (P.V. of Q). R divides PQ internally in ratio 2:1.
By the section formula, the position vector of R is:
rˉ=2+12qˉ+1⋅pˉ
=32(i^+4j^−2k^)+(i^−2j^+k^)
=3(2i^+8j^−4k^)+(i^−2j^+k^)=33i^+6j^−3k^
=i^+2j^−k^
✓Final answerPosition vector of R =i^+2j^−k^
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