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Finding Components of a vector Along and Perpendicular to another Vector - Practice Questions & MCQ

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

Syllabus

Quick Facts

Finding Components of a vector Along and Perpendicular to another Vector is considered one the most difficult concept.
7 Questions around this concept.

Solve by difficulty

EASY

Let ABCD be a parallelogram such that $\underset{AB}{\rightarrow}$ $= \vec{q}$ , $\underset{AD}{\rightarrow}$ $= \vec{p}$ and $\angle BAD$ be an acute angle. If $\vec{r}$ is the vector that coincides with the altitude directed from the vertex B to the side AD, then $\vec{r}$ is given by

$\vec{r}= 3\vec{q}-\frac{3\left ( \vec{p}.\vec{q} \right )}{\left ( \vec{p}.\vec{p} \right )}\vec{p}$

$\vec{r}= -\vec{q}+\left ( \frac{\vec{p}.\vec{q}}{\vec{p}.\vec{p}} \right )\vec{p}$

$\vec{r}= \vec{q}-\left ( \frac{\vec{p}.\vec{q}}{\vec{p}.\vec{p}} \right )\vec{p}$

$\vec{r}= -3\vec{q}+\frac{3\left ( \vec{p}.\vec{q} \right )}{\left ( \vec{p}.\vec{p} \right )}\vec{p}$

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Concepts Covered - 1

Finding Components of a vector Along and Perpendicular to another Vector

Let $\vec{a}$ and $\vec{b}$ be two vectors represented by $\overrightarrow{O A}$ and $\overrightarrow{O B}$ respectively and let Ө be the angle between $\vec{a}$ and $\vec{b}$ . Then,

$\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\vec{b}=\overrightarrow{O M}+\overrightarrow{M B}$

$\\\text{Also,}\;\;\;\;\;\;\;\;\;\overrightarrow{O M}=(O M) \hat{a}\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}=(O B \cos \theta) \hat{a}\;=\;=(|\vec{b}| \cos \theta) \hat{a}\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}= \left ( \frac{\left ( \vec a\cdot \vec b \right )}{|\vec a|} \right )\hat a=\left ( \frac{\left ( \vec a\cdot \vec b \right )}{|\vec a||\vec a|} \right )\vec a\;\;\;\;\;\;\;\;\;\;\;\;\;\left [ \because \hat a=\frac{\vec a}{|\vec a|} \right ]\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}= \left ( \frac{\left ( \vec a\cdot \vec b \right )}{|\vec a|^2} \right )\vec a$

$\\\mathrm{As,\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\vec{b}=\overrightarrow{O M}+\overrightarrow{M B}\\\mathrm{\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;}\overrightarrow{M B}=\vec{b}-\overrightarrow{O M}=\vec b-\left ( \frac{\left ( \vec a\cdot \vec b \right )}{|\vec a|^2} \right )\vec a\\\text {Thus, the components of } \vec{b} \text { along and perpendicular to } \vec{a} \text { are }\;\left ( \frac{\left ( \vec a\cdot \vec b \right )}{|\vec a|^2} \right )\vec a\\\text{and }\;\vec b-\left ( \frac{\left ( \vec a\cdot \vec b \right )}{|\vec a|^2} \right )\vec a,\;\text{ respectively.}$

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Finding Components of a vector Along and Perpendicular to another Vector

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