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Let \vec{a},\vec{b} and \vec{c} be three unit vectors such that \vec{a}+\vec{b}+\vec{c}=\vec{0}.If \lambda =\vec{a}\cdot \vec{b}+\vec{b}\cdot \vec{c}+\vec{c}\cdot \vec{a} and \vec{d} =\vec{a}\times \vec{b}+\vec{b}\times \vec{c}+\vec{c}\times \vec{a}, then the ordered pair,  \left ( \lambda ,\vec{d} \right ) is equal to :
Option: 1 \left [ \frac{3}{2},3\vec{a}\times \vec{c} \right ]      
Option:2  \left [ -\frac{3}{2},3\vec{c}\times \vec{b} \right ]

Option: 3 \left [ -\frac{3}{2},3\vec{a}\times \vec{b} \right ]

Option: 4  \left [ \frac{3}{2},3\vec{b}\times \vec{c} \right ]

Answers (1)




Addition and subtraction of Vectors -


Properties of vector addition

The sum of two vectors is always a vector.

\\1.\;\;\;\;\vec a+\vec b =\vec b +\vec a \quad \quad\quad\;\;\;\;\;\;\;\;\;\;\;\;\; \text { (Commutative property) }\\2.\;\;\;(\vec{a}+\vec{b})+\vec{c}=\vec{a}+(\vec{b}+\vec{c})\quad \quad \text { (Associative property) }\\3.\;\;\;\;\vec{a}+\overrightarrow{0}=\overrightarrow{0}+\vec{a}=\vec{a}\quad \quad\quad\;\; \text { (additive identity) }\\4.\;\;\;\;\vec{a}+{\left (-\vec a \right )}={\left ( -\vec a \right )}+\vec{a}=\vec{0}\quad \quad \text { (additive inverse) }


Properties of vector Subtraction

\\1.\;\;\;\;\vec a-\vec b\neq \vec b-\vec a\\2.\;\;\;\;(\vec a-\vec b )-\vec c\neq \vec a-(\vec b-\vec c)\\3.\;\;\;\;\mathrm{For \;any \;two\;vectors\;\;\ \overrightarrow{a}\;\;and\;\; \overrightarrow{b}}\\\mathrm{\;\;\;\;\;\;\;\;\;\;(a)}\;\; {|a+b| \leq|a|+|b|} \\\mathrm{\;\;\;\;\;\;\;\;\;\;(b)}\;\; {|a+b| \geq|a|-|b|} \\\mathrmlk{\;\;\;\;\;\;\;\;\;\;(c)}\;\; {|a-b| \leq|a|+|b|} \\\mathrm{\;\;\;\;\;\;\;\;\;\;(d)}\;\; {|a-b| \geq|a|-|b|}


{|\vec{a}+\vec{b}+\vec{c}|^{2}=0} \\ {3+2(\vec{a} \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a})=0} \\ {(\vec{a} \cdot \vec{b}+\vec{b} \cdot \vec{c}+\vec{c} \cdot \vec{a})=\frac{-3}{2}} \\ {\Rightarrow \lambda=\frac{-3}{2}}


\\ {\vec{d}=\vec{a} \times \vec{b}+\vec{b} \times(-\vec{a}-\vec{b})+(-\vec{a}-\vec{b}) \times \vec{a}} \\ {=\vec{a} \times \vec{b}+\vec{a} \times \vec{b}+\vec{a} \times \vec{b}} \\ {\vec{d}=3(\vec{a} \times \vec{b})}

Correct Option (3)

Posted by

Ritika Jonwal

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