Question

If $P_1\left(x_1,y_1,z_1\right)$ and $P_2\left(x_2,y_2,z_2\right)$ are any two points, and the vector joining $P_1$ and $P_2$ is the vector $P_1{P}_2$, then

• A. $P_1{P}_2=O{P}_2-O{P}_1$
• B. $P_1{P}_2=\left(x_2-x_1\right)\hat{i}+\left(y_2-y_1\right)\hat{j}+\left(z_2-z_1\right)\hat{k}$
• C. $\left|P_1{P}_2\right|=\sqrt{{\left(x_2-x_1\right)}^2+(y_2-y_{1)}^2+{\left(z_2-z_1\right)}^2}$
• D. All of the above

Answer 1

Answer: (D)

If $P_1\left(x_1,y_1,z_1\right)$ and $P_2\left(x_2,y_2,z_2\right)$ are any two points, the the vector joining $P_1$ and $P_2$ is the vector $P_1{P}_2$.

Joining the points $P_1$ and $P_2$ with the origin $O$ and applying triangle law, from the triangle $O{P}_1{P}_2$ we have
$O{P}_1+P_1{P}_2=O{P}_2$
Using the properties of vector addition, the above equation becomes
$P_1{P}_2=O{P}_2-O{P}_1$
i.e., $P_1{P}_2=\left(x_2\hat{i}+y_2\hat{j}+z_2\hat{k}\right)-\left(x_1\hat{i}+y_1\hat{j}+z_1\hat{k}\right)$
$=\left(x_2-x_1\right)\hat{i}+\left(y_2-y_1\right)\hat{j}+\left(z_2-z_1\right)\hat{k}$
The magnitude of vector $P_1{P}_2$ is given by
$P_1{P}_2=\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2+{\left(z_2-z_1\right)}^2}$
So, option (d) is correct.