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,  $\overrightarrow{O A}, \overrightarrow{O B}$ and $\overrightarrow{O C}$ each having magnitude 1, are called unit vectors along the axes $O X$, $O Y$, and $O Z$, 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{aligned}
& \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{aligned}
$
and,
Hence, the position vector of P with reference to O is given by
$
\overrightarrow{\mathrm{OP}}(\text { or } \vec{r})=x \hat{i}+y \hat{j}+z \hat{k}
$

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

Vector Joining Two Points

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

Applying the triangle law of addition on the triangle $O A B$
$
\overrightarrow{O A}+\overrightarrow{A B}=\overrightarrow{O B}
$

Using the properties of vector addition, the above equation becomes
$
\overrightarrow{A B}=\overrightarrow{O B}-\overrightarrow{O A}
$
i.e.
$
\begin{aligned}
\overrightarrow{A B} & =\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}
\end{aligned}
$

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