If the position vectors of three consecutive vertices, of a parallelogram are veci+ vecj+ veck, veci+3 vecj+5 veck and 7 veci+9 vecj+11 veck, then the coordinates of the fourth vertex are

Question

If the position vectors of three consecutive vertices, of a parallelogram are veci+ vecj+ veck, veci+3 vecj+5 veck and 7 veci+9 vecj+11 veck, then the coordinates of the fourth vertex are (A) (2, 1, 3) (B) (6, 7, 8) (C) (4, 1, 3) (D) (7, 7, 7)

Tardigrade Admin · Accepted Answer

Let the vertices of a parallelogram are A(1, 1, 1) B(1, 3, 5), C(7, 9, 11) and fourth vertex be D (

x

, y, z) Midpoint of AC is (4, 5, 6) and midpoint of BD is

(\frac{1 + x}{2}, \frac{3 + y}{2}, \frac{5 + z}{2})

.
In a parallelogram midpoint of diagonals are coincide.

∴

\frac{1 + x}{2} = 4, \frac{3 + y}{2} = 5, \frac{5 + z}{2} = 6

\Rightarrow

x = 7, y = 7, z = 7

Q. If the position vectors of three consecutive vertices, of a parallelogram are $i + j + k,$ $i + 3 j + 5 k$ and $7 i + 9 j + 11 k,$ then the coordinates of the fourth vertex are

$(2, 1, 3)$

$(6, 7, 8)$

$(4, 1, 3)$

$(7, 7, 7)$

$(8, 8, 8)$

Q. If the position vectors of three consecutive vertices, of a parallelogram are i+j​+k, i+3j​+5k and 7i+9j​+11k, then the coordinates of the fourth vertex are

(2,1,3)

(6,7,8)

(4,1,3)

(7,7,7)

(8,8,8)