JEE Main 2023 · 15 April · morning · Single correct · +4 / −1

Let ABCD be a quadrilateral. If E and F are the mid points of the diagonals AC and BD respectively and (A B⃗-B C⃗)+(A D⃗-D C⃗)=k F E⃗, then k is equal to :

  1. (a) -2
  2. (b) 4
  3. (c) -4
  4. (d) 2
Answer and solution

Answer (c) -4

Solution

Maths sir · 3 steps

Turn every side into position vectors; the sums a⃗ + c⃗ and b⃗ + d⃗ become the midpoints.

  1. Set up. Let A, B, C, D have position vectors a⃗, b⃗, c⃗, d⃗. Then e⃗ = (a⃗ + c⃗)/2 and f⃗ = (b⃗ + d⃗)/2, so a⃗ + c⃗ = 2e⃗ and b⃗ + d⃗ = 2f⃗.
  2. Expand each bracket. AB⃗ - BC⃗ = (b⃗ - a⃗) - (c⃗ - b⃗) = 2b⃗ - a⃗ - c⃗. In the same way, AD⃗ - DC⃗ = 2d⃗ - a⃗ - c⃗.

    2(b⃗ + d⃗) - 2(a⃗ + c⃗) = 4f⃗ - 4e⃗ = -4(e⃗ - f⃗) = -4 FE⃗

  3. So. Since FE⃗ = e⃗ - f⃗, the sum equals -4 FE⃗, which gives k = -4. The answer is (c).

The idea to keep. For quadrilateral vector questions, write every side as head minus tail in position vectors. Then a⃗ + c⃗ and b⃗ + d⃗ become twice the diagonal midpoints.

Answer from the NTA final answer key. Solution by PrepWiser.

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Asked about 220 times in JEE Main since 2020. This question is from 2023.