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Q.
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
Vector Algebra
Solution:
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 $OP _1 P _2$ we have
$OP _1+ P _1 P _2= OP _2$
Using the properties of vector addition, the above equation becomes
$P _1 P_2 = OP _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.
