NDA 2 2025 Maths, question 69

Topic: Vectors

Q69

The position vectors of the vertices $A,B,C$ and $D$ of a quadrilateral $ABCD$ are given by $3\hat{i}+4\hat{j}-2\hat{k}$ , $4\hat{i}-4\hat{j}-3\hat{k}$ , $2\hat{i}-3\hat{j}+2\hat{k}$ and $6\hat{i}-2\hat{j}+\hat{k}$ respectively. What is the angle between the diagonals $AC$ and $BD$ of the quadrilateral?

(a) $90^{\circ}$ Correct answer
(b) $75^{\circ}$
(c) $60^{\circ}$
(d) $45^{\circ}$

Solution.

Answer: (a) $90^{\circ}$

AC=C−A=(−1,−7,4) and BD=D−B=(2,2,4).
Their dot product is −2−14+16=0.
Both are nonzero, so the diagonals are perpendicular and the angle is 90°.