Vectors Joining Two Points - Practice Questions & MCQ

Let the points A(1, 0, 0), B(0, 1, 0) and C(0, 0, 1) on the x-axis, y-axis and z-axis, respectively. Then, clearly,$|\overrightarrow{\mathrm{OA}}|=1,|\overrightarrow{\mathrm{OB}}|=1$ and $|\overrightarrow{\mathrm{OC}}|=1$

The vectors,  $\overrightarrow{OA},\overrightarrow{OB}$ and $\overrightarrow{OC}$ each having magnitude 1, are called unit vectors along the axes $OX$, $OY$, and $OZ$, respectively, and denoted by $\hat{\mathbf{i}},\hat{\mathbf{j}}$, and $\hat{\mathbf{k}}$ respectively.
Now consider any point $\mathrm{P}(\mathrm{x},\mathrm{y},\mathrm{z})$ with position vector OP . Let ${\mathrm{P}}_1$ be the foot of the perpendicular from P on the plane XOY . As we observe that $P_{1}P$ is parallel to the z-axis. Also, $\hat{\mathbf{i}},\hat{\mathbf{j}}$, and $\hat{\mathbf{k}}$ are the unit vectors along the $x,y$, and $z$-axes, respectively, thus,  by the definition of the coordinates of P, we have $\overrightarrow{{\mathrm{P}}_1\mathrm{P}}=\overrightarrow{\mathrm{OR}}=z\hat{\mathbf{k}}$.

Similarly, $\overrightarrow{{\mathrm{QP}}_1}=\overrightarrow{\mathrm{OS}}=y\hat{\mathbf{j}}$ and $\overrightarrow{\mathrm{OQ}}=x\hat{\mathbf{i}}$

Therefore,
$\begin{array}{rl}& \overrightarrow{{\mathrm{OP}}_1}=\overrightarrow{\mathrm{OQ}}+\overrightarrow{{\mathrm{QP}}_1}=x\hat{i}+y\hat{j}\\ & \overrightarrow{\mathrm{OP}}=\overrightarrow{{\mathrm{OP}}_1}+\overrightarrow{{\mathrm{P}}_1\mathrm{P}}=x\hat{i}+y\hat{j}+z\hat{k}\end{array}$
and,
Hence, the position vector of P with reference to O is given by
$\overrightarrow{\mathrm{OP}}(\text{ or }\overrightarrow{r})=x\hat{i}+y\hat{j}+z\hat{k}$

And, the length of any vector $\overrightarrow{r}=x\hat{i}+y\hat{j}+z\hat{k}$ is given by
$|\overrightarrow{r}|=|x\hat{i}+y\hat{j}+z\hat{k}|=\sqrt{x^2+y^2+z^2}$

Vector Joining Two Points

If $\mathrm{A}({\mathrm{x}}_1,{\mathrm{y}}_1,{\mathrm{z}}_1)$ and $\mathrm{B}({\mathrm{x}}_2,{\mathrm{y}}_2,{\mathrm{z}}_2)$ are any two points in three - dimensional system, then vector joining point A and B is the vector $\overrightarrow{AB}$.
Joining the point A and B with the origin, O , we get position vector of point A and B . i.e.
$\begin{array}{rl}& \overrightarrow{OA}=x_1\hat{i}+y_1\hat{j}+z_1\hat{k}\\ & \overrightarrow{OB}=x_2\hat{i}+y_2\hat{j}+z_2\hat{k}\end{array}$

Applying the triangle law of addition on the triangle $OAB$
$\overrightarrow{OA}+\overrightarrow{AB}=\overrightarrow{OB}$

Using the properties of vector addition, the above equation becomes
$\overrightarrow{AB}=\overrightarrow{OB}-\overrightarrow{OA}$
i.e.
$\begin{array}{rl}\overrightarrow{AB}& =(x_2\hat{i}+y_2\hat{j}+z_2\hat{k})-(x_1\hat{i}+y_1\hat{j}+z_1\hat{k})\\ & =(x_2-x_1)\hat{i}+(y_2-y_1)\hat{j}+(z_2-z_1)\hat{k}\end{array}$

The magnitude of vector $\overrightarrow{AB}$ is given by
$|\overrightarrow{AB}|=\sqrt{{({\mathrm{x}}_2-{\mathrm{x}}_1)}^2+{({\mathrm{y}}_2-{\mathrm{y}}_1)}^2+{({\mathrm{z}}_2-{\mathrm{z}}_1)}^2}$