The position vectors of three consecutive vertices A, B and C of a parallelogram ABCD are r1,r2, and r3, respectively. Then the position vector of the fourth vertex D is

Question

The position vectors of three consecutive vertices A, B and C of a parallelogram ABCD are r1,r2, and r3, respectively. Then the position vector of the fourth vertex D is (A) r1+r2-r3  (B) r2+r3-r1  (C) r3+r1-r2  (D) none of these.

Tardigrade Admin · Accepted Answer

Since the diagonals of a parallelogram bisect each other ∴ ( vecr1+ vecr3/2)=( vecr2+ vecr4/2) = vecr1+ vecr3= vecr2+ vecr4 ∴ vecr4= vecr3+ vecr1- vecr2, where D is ( vecr4)

Q. The position vectors of three consecutive vertices A, B and C of a parallelogram ABCD are $r_{1}, r_{2},$ and $r_{3},$ respectively. Then the position vector of the fourth vertex D is

$r_{1} + r_{2} - r_{3}$

$r_{2} + r_{3} - r_{1}$

$r_{3} + r_{1} - r_{2}$

none of these.

Since the diagonals of a parallelogram bisect each other
$∴ \frac{r _{1} + r _{3}}{2} = \frac{r _{2} + r _{4}}{2}$
$= r_{1} + r_{3} = r_{2} + r_{4}$
$∴ r_{4} = r_{3} + r_{1} - r_{2}$ , where $D$ is $(r_{4})$

Q. The position vectors of three consecutive vertices A, B and C of a parallelogram ABCD are r1​​,r2​​, and r3​​, respectively. Then the position vector of the fourth vertex D is

r1​​+r2​​−r3​​

r2​​+r3​​−r1​​

r3​​+r1​​−r2​​