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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

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.

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QuestionEASY

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

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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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