Tag: addition of vectors

Questions Related to addition of vectors

If $\overrightarrow A ,\overrightarrow B $ and $\overrightarrow C $ are vectors such that $\left| {\overrightarrow B } \right| = \left| {\overrightarrow C } \right|$ , then  $\left{ {\left( {\overrightarrow A  + \overrightarrow B } \right)} \right. \times \left. {\left( {\overrightarrow A  + \overrightarrow C } \right)} \right} \times \left( {\overrightarrow B  \times \overrightarrow C } \right).\left( {\overrightarrow B  + \overrightarrow C } \right) = 1 $  these relation is ?

maths addition of vectors vectors from a geometric viewpoint basic concepts of vector introduction to vectors

  1. True

  2. False

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Correct Option: B

If the vectors $\overrightarrow a  = \left( {2,{{\log } _3}x,\;a} \right)$ $and\;\overrightarrow b  = \left( { - 3,a{{\log } _3}x,{{\log } _3}x} \right)$ are included at an acute angle then-

maths addition of vectors vectors from a geometric viewpoint basic concepts of vector introduction to vectors

  1. a=0

  2. a<0

  3. a>0

  4. None of these

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Correct Option: C
Explanation:

$\begin{array}{l} \overrightarrow { a } =2\left( { { { \log   } _{ 3 } }x,a } \right) ,\overrightarrow { b } =\left( { -3,a{ { \log   } _{ 3 } }x,{ { \log   } _{ 3 } }x } \right)  \ \overrightarrow { a } \overrightarrow { b } =\left| a \right| \left| b \right| \cos  \theta  \ \cos  \theta =\frac { { \overrightarrow { a } .\overrightarrow { b }  } }{ { \left| a \right| \left| b \right|  } }  \ =\frac { { -6+a{ { \left( { { { \log   } _{ 3 } }x } \right)  }^{ 2 } }+a.{ { \log   } _{ 3 } }x } }{ { \sqrt { 4+{ { \left( { { { \log   } _{ 3 } }x } \right)  }^{ 2 } }+{ a^{ 2 } } } .\sqrt { 9+{ a^{ 2 } }{ { \left( { { { \log   } _{ 3 } }x } \right)  }^{ 2 } }+{ { \left( { { { \log   } _{ 3 } }x } \right)  }^{ 2 } } }  } }  \ for\, \, acute\, \, angle\, \, a>0 \ Option\, \, \, C\, \, is\, \, correct. \end{array}$

If the vector $\bar{c}, \bar{a} = x\bar{i}+y\bar{j}+ z\bar{k}, \bar{b}= \bar{j}$ are such that $\bar{a}, \bar{c}, \bar{b}$ from R.H.S then $\bar{c}$ = 

maths addition of vectors vectors from a geometric viewpoint basic concepts of vector introduction to vectors

  1. $z\bar{i} -x\bar{k}$

  2. $z\bar{i} -3\bar{k}$

  3. $x\bar{j} -y\bar{k}$

  4. $y\bar{j} -x\bar{k}$

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Correct Option: A

If $a,b,c$ are unit vectors, then the maximum value of $|a+2b|^{2}+|b+3c|^{2}+|c+4a|^{2}$ is 

maths addition of vectors vectors from a geometric viewpoint basic concepts of vector introduction to vectors

  1. $50$

  2. $21$

  3. $48$

  4. $58$

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Correct Option: A
Explanation:
Given,

$\left|a+2b\right|^2+\left|b+3c\right|^2+\left|c+4a\right|^2$

$=\left(a+2b\right)^2+\left(b+3c\right)^2+\left(4a+c\right)^2$

$=a^2+4ab+4b^2+b^2+6bc+9c^2+c^2+8ac+16a^2$

$=17a^2+4ab+8ac+5b^2+10c^2+6bc$

$=50$