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Question: If the position vectors of the point A, B, C be **i, j, k** respectively and P be a point such that ...

If the position vectors of the point A, B, C be i, j, k respectively and P be a point such that AB→=CP→,\overset{\rightarrow}{AB} = \overset{\rightarrow}{CP}, then the position vector of P is

A

−i+j+k- \mathbf{i} + \mathbf{j} + \mathbf{k}

B

−i−j+k- \mathbf{i} - \mathbf{j} + \mathbf{k}

C

i+j−k\mathbf{i} + \mathbf{j} - \mathbf{k}

D

None of these

Answer

−i+j+k- \mathbf{i} + \mathbf{j} + \mathbf{k}

Explanation

Solution

Let the position vector of P is xi+yj+zk,x\mathbf{i} + y\mathbf{j} + z\mathbf{k},

then AB→=CP→⇒j−i=xi+yj+(z−1)k\overset{\rightarrow}{AB} = \overset{\rightarrow}{CP} \Rightarrow \mathbf{j} - \mathbf{i} = x\mathbf{i} + y\mathbf{j} + (z - 1)\mathbf{k}

By comparing the coefficients of i,j\mathbf{i},\mathbf{j} and k,\mathbf{k}, we get x=−1,x = - 1,

y=1andz–1=0⇒z=1y = 1\text{andz}\text{–}\text{1} = 0 \Rightarrow z = 1

Hence required position vector is −i+j+k.- \mathbf{i} + \mathbf{j} + \mathbf{k}.