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Vectors Joining Two Points - Practice Questions & MCQ

Edited By admin | Updated on Sep 18, 2023 18:34 AM | #JEE Main

Quick Facts

Component of vector and Vector Joining Two Points is considered one of the most asked concept.
7 Questions around this concept.

Concepts Covered - 1

Component of vector and Vector Joining Two Points

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,
$|\overrightarrow{\mathrm{OA}}|=1,|\overrightarrow{\mathrm{OB}}|=1 \text { and }|\overrightarrow{\mathrm{OC}}|=1$

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 P(x, y, z) with position vector OP. Let P₁ be the foot of the perpendicular from P on the plane XOY. As, we observe that P₁P 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, $\overrightarrow{\mathrm{QP}_{1}}=\overrightarrow{\mathrm{OS}}=y \hat{\mathbf j} \text{ and } \overrightarrow{\mathrm{OQ}}=x \hat{\mathbf i}$

$\\\text{Therefore,}\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;} {\overrightarrow{\mathrm{OP}_{1}}=\overrightarrow{\mathrm{OQ}}+\overrightarrow{\mathrm{QP}_{1}}=x \hat{i}+y \hat{j}} \\ \\\mathrm{and,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;} {\overrightarrow{\mathrm{OP}}=\overrightarrow{\mathrm{OP}_{1}}+\overrightarrow{\mathrm{P}_{1} \mathrm{P}}=x \hat{i}+y \hat{j}+z \hat{k}}\\\\\text{Hence, the position vector of P with reference to O is given by}\\\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;} \overrightarrow{\mathrm{OP}}(\text { or } \vec{r})=x \hat{i}+y \hat{j}+z \hat{k}\\\\\text {And, the length of any vector } \vec{r}=x \hat{i}+y \hat{j}+z \hat{k} \text { is given by }\\\\\mathrm{\;\;\;\;\;\;\;\;}|\vec{r}|=|x \hat{i}+y \hat{j}+z \hat{k}|=\sqrt{x^{2}+y^{2}+z^{2}}$

Vector Joining Two Points

$\\\mathrm{If \;A(x_1,y_1,z_1)\;and\;B(x_2,y_2,z_2)\;are\; any\; two\; points\;in\;three-dimensional\;system,\;then}\\\mathrm{vector\;joining\;point\;A\;and\;B\;is\;the\;vector\;\overrightarrow{AB}.}\\\mathrm{Joining\;the\;point \;A\;and\;B\;with\;the\;origin,\;O,\;we\;get\;position\;vector\;of\;point\;A\;and\;B.}\\\text{i.e.}\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;}\overrightarrow{OA}=x_1\hat i+y_1\hat j+z_1\hat k\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;}\overrightarrow{OB}=x_2\hat i+y_2\hat j+z_2\hat k\\\text{Applying triangle law of addition on the triangle OAB}\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;}\overrightarrow{OA}+\overrightarrow{AB}=\overrightarrow{OB}\\\text{Using the properties of vector addition, the above equation becomes}\\\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;}\overrightarrow{AB}=\overrightarrow{OB}-\overrightarrow{OA}$
$\\\text{i.e.}\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;}\overrightarrow{AB}=\left(x_{2} \hat{i}+y_{2} \hat{j}+z_{2} \hat{k}\right)-\left(x_{1} \hat{i}+y_{1} \hat{j}+z_{1} \hat{k}\right)\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}=\left(x_{2}-x_{1}\right) \hat{i}+\left(y_{2}-y_{1}\right) \hat{j}+\left(z_{2}-z_{1}\right) \hat{k}\\\\\text { The magnitude of vector } \overrightarrow{\mathrm{A} \mathrm{B}} \text { is given by }\\\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;|\overrightarrow{AB}|=\sqrt{\left(x_{2}-x_{1}\right)^{2}+\left(y_{2}-y_{1}\right)^{2}+\left(z_{2}-z_{1}\right)^{2}}}$