Question

If ABCD is a parallelogram and the position vectors of A, B, C are $\mathbf{i} + 3\mathbf{j} + 5\mathbf{k},\mathbf{i} + \mathbf{j} + \mathbf{k}$ and $7\mathbf{i} + 7\mathbf{j} + 7\mathbf{k},$ then the position vector of D will be

• A. $7\mathbf{i} + 5\mathbf{j} + 3\mathbf{k}$
• B. $7\mathbf{i} + 9\mathbf{j} + 11\mathbf{k}$
• C. $9\mathbf{i} + 11\mathbf{j} + 13\mathbf{k}$
• D. $8\mathbf{i} + 8\mathbf{j} + 8\mathbf{k}$

Answer 1

Answer: (B)

$7\mathbf{i} + 9\mathbf{j} + 11\mathbf{k}$

Answer 2

Let position vector of D is $x\mathbf{i} + y\mathbf{j} + z\mathbf{k},$ then $\overset{\rightarrow}{AB} = \overset{\rightarrow}{DC}$ $\Rightarrow - 2\mathbf{j} - 4\mathbf{k} = (7 - x)\mathbf{i} + (7 - y)\mathbf{j} + (7 - z)\mathbf{k}$

$\Rightarrow x = 7,y = 9,z = 11.$

Hence position vector of $D$ will be $7\mathbf{i} + 9\mathbf{j} + 11\mathbf{k}$.