Exercise 10.4
Let the vectors \[\overrightarrow{a}\] and \[\overrightarrow{b}\] be such that \[\left| \overrightarrow{a} \right|=3\] and \[\left| \overrightarrow{b} \right|=\frac{\sqrt{2}}{3}\], then \[\overrightarrow{a}\times \overrightarrow{b}\] is a unit vector, if the angle between \[\overrightarrow{a}\] and \[\overrightarrow{b}\] is (A) \[\frac{\pi }{6}\] (B) \[\frac{\pi }{4}\] (C) \[\frac{\pi }{3}\] (D) \[\frac{\pi }{2}\]
Find the area of the parallelogram whose adjacent sides are determined by the vector .\[\overrightarrow{a}=\widehat{i}-\widehat{j}+3\widehat{k}\] and \[\overrightarrow{b}=2\widehat{i}-7\widehat{j}+\widehat{k}\]
Let us consider,
Find the area of the triangle with vertices A \[(1,1,2)\], B \[(2,3,5)\] and C \[(1,5,5)\].
We know that,
If either \[\overrightarrow{a}=0\] or \[\overrightarrow{b}=0\] then \[\overrightarrow{a}\times \overrightarrow{b}=0\]. Is the converse true? Justify your answer with an example.
Firstly let us consider,
Let the vectors \[\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}\] given as \[{{a}_{1}}\widehat{i}+{{a}_{2}}\widehat{j}+{{a}_{3}}\widehat{k}\],\[{{b}_{1}}\widehat{i}+{{b}_{2}}\widehat{j}+{{b}_{3}}\widehat{k}\],\[{{c}_{1}}\widehat{i}+{{c}_{2}}\widehat{j}+{{c}_{3}}\widehat{k}\] . Then show that \[\overrightarrow{a}\times (\overrightarrow{b}+\overrightarrow{c})=\overrightarrow{a}\times \overrightarrow{b}+\overrightarrow{a}\times \overrightarrow{c}\]
It is given that,
Given that \[\overrightarrow{a}.\overrightarrow{b}=0\] and \[\overrightarrow{a}\times \overrightarrow{b}=0\]. What can you conclude about the vectors \[\overrightarrow{a}\] and \[\overrightarrow{b}\] ?
It is given that
Find \[\lambda \] and \[\mu \] if \[(2\widehat{i}+6\widehat{j}+27\widehat{k})\times (\widehat{i}+\lambda \widehat{j}+\mu \widehat{k})=\overrightarrow{0}\].
It is given that,
Show that \[(\overrightarrow{a}-\overrightarrow{b})\times (\overrightarrow{a}+\overrightarrow{b})=2(\overrightarrow{a}\times \overrightarrow{b})\]
Firstly consider the LHS, We have,
If a unit vector \[\overrightarrow{a}\] makes an angles \[\frac{\pi }{3}\] with \[\widehat{i},\frac{\pi }{4}\] with \[\widehat{j}\]and an acute angle \[\theta \] with \[\widehat{k}\] , then find \[\theta \] and hence, the compound of \[\overrightarrow{a}\]
Firstly,
Find a unit vector perpendicular to each of the vector \[\overrightarrow{a}+\overrightarrow{b}\] and \[\overrightarrow{a}-\overrightarrow{b}\], where \[\overrightarrow{a}=3\widehat{i}+2\widehat{j}+2\widehat{k}\] and \[\overrightarrow{b}=\widehat{i}+2\widehat{j}-2\widehat{k}\].
It is given that:
Find \[\left| \overrightarrow{a}\times \overrightarrow{b} \right|\], if \[\overrightarrow{a}=\widehat{i}-7\widehat{j}+7\widehat{k}\] and \[\overrightarrow{b}=3\widehat{i}-2\widehat{j}+2\widehat{k}\]
It is given that: