Tag: introduction to vector algebra

Questions Related to introduction to vector algebra

Which of the following expressions represent vectors ?

maths vectors and transformations introduction to vector algebra algebra of vectors operations on vectors

  1. $\displaystyle -v$

  2. $\displaystyle u+v$

  3. $\displaystyle \vec{u}+ \vec{v} $

  4. $\displaystyle -\left| \vec { A  B} \right| $

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Correct Option: C
Explanation:

$\vec u + \vec v$ represents vector.


Hence, the option $C$ is correct.

State the following statement is true or false

In vector Algebra displacement is a vector quantity.

maths vectors and transformations introduction to vector algebra algebra of vectors operations on vectors

  1. True

  2. False

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Correct Option: A
Explanation:
In vector algebra, displacemant is a vector quantity. since displacement has both value and direction. 
Distance is a scalar quantity.
Therefore, given statement is $TRUE$. 

On ethe rectangularcomponent of a viewer of 20m is 10 m .The component is

maths vectors and transformations introduction to vector algebra algebra of vectors operations on vectors

  1. 20m

  2. $\eqalign{
    & \cr
    & 20\sqrt 3 \cr} $

  3. $\eqalign{
    & \cr
    & 10\sqrt 3 \cr} $

  4. None

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Correct Option: A

$[ a \times b \times c ] =  2[ a b c ] ^ { 2 }$

maths vectors and transformations introduction to vector algebra algebra of vectors operations on vectors

  1. True

  2. False

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Correct Option: B

If the position vectors of the vertices of atriangle are $2 \overline { i } - \overline { j } + \overline { k } , \overline { i } - 3 \vec { j } - 5 \overline { k }$ and $3 \vec { i } - 4 \overline { j } - 4 \overline { k }$ then the triangle is

maths vectors and transformations introduction to vector algebra algebra of vectors operations on vectors

  1. Equilateral triangle

  2. Isosceles triangle

  3. Right angled isosceles triangle

  4. Right angled triangle

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Correct Option: D
Explanation:

$\\Let\>A\>(2\hat{i}-\hat{j}+\hat{k}),\>B(\hat{i}-3\hat{j}-5\hat{k})\>and\\C(3\hat{i}-4\hat{j}-4\hat{k})\\then\\\overrightarrow{AB}=-\hat{i}-2\hat{j}-6\hat{k}\\\therefore\>|\overrightarrow{AB}|=\sqrt{1+4+36}=\sqrt{41}\\\overrightarrow{BC}=2\hat{i}-\hat{j}+\hat{k}\\\therefore\>|\overrightarrow{BC}|=\sqrt{4+1+1}=\sqrt{6}\\\overrightarrow{CA}=-\hat{i}+3\hat{j}+5\hat{k}\\\therefore\>|\overrightarrow{CA}|=\sqrt{1+9+25}=\sqrt{35}\\clearly\>\>\>|\overrightarrow{BC}|^2+|\overrightarrow{CA}|^2=|\overrightarrow{AB}|^2\\\therefore\>Triangle\>is\>a\>right\>angled\>triangle$

If $\displaystyle {\sec}^{2}A\hat{i}+\hat{j}+\hat{k}$, $\displaystyle \hat{i}+{\sec}^{2}B\hat{j}+\hat{k}$,and $\displaystyle \hat{i}+\hat{j}+{\sec}^{2}C\hat{k}$, are coplanar then $\displaystyle {\cot}^{2}A+{\cot}^{2}B+{\cot}^{2}{C}$ is    

maths vectors and transformations introduction to vector algebra algebra of vectors operations on vectors

  1. $1$

  2. $2$

  3. $0$

  4. $-1$

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Correct Option: D
Explanation:

$\begin{array}{l} \left| { \begin{array} { *{ 20 }{ c } }{ { { \sec   }^{ 2 } }A } & 1 & 1 \ 1 & { { { \sec   }^{ 2 } }B } & 1 \ 1 & 1 & { { { \sec   }^{ 2 } }C } \end{array} } \right| =0 \\ { C _{ 1 } }\to { C _{ 1 } }-{ C _{ 2 } }\, \, \, \, \, \, \, \, \, { C _{ 2 } }\to { C _{ 2 } }-{ C _{ 3 } } \\ \left| { \begin{array} { *{ 20 }{ c } }{ { { \tan   }^{ 2 } }A } & 0 & 1 \ { -{ { \tan   }^{ 2 } }B } & { { { \tan   }^{ 2 } }B } & 1 \ 0 & { -{ { \tan   }^{ 2 } }C } & { { { \sec   }^{ 2 } }C } \end{array} } \right| =0 \\ { \tan ^{ 2 }  }A\left[ { { { \tan   }^{ 2 } }B{ { \sec   }^{ 2 } }C+{ { \tan   }^{ 2 } }C } \right] +{ \tan ^{ 2 }  }B{ \tan ^{ 2 }  }C=0 \ \\dfrac { { { { \sec   }^{ 2 } }C } }{ { { { \tan   }^{ 2 } }C } } +{ \cot ^{ 2 }  }B+{ \cot ^{ 2 }  }A=0 \ \\cos  e{ c^{ 2 } }C+{ \cot ^{ 2 }  }B+{ \cot ^{ 2 }  }A=0 \\ { \cot ^{ 2 }  }A+{ \cot ^{ 2 }  }B+{ \cot ^{ 2 }  }C=-1 \\ Hence,\, the\, option\, D\, is\, \, the\, correct\, answer. \end{array}$

Let        $\dot{a}$ = $\hat{i}$ + $\hat{j}$ + $\sqrt{2}\hat{k}$
              $\dot{b}$ = $b _1\hat{i}$ + $b _2\hat{j}$ + $\sqrt{2}\hat{k}$
              $\dot{c}$ = $5\hat{i}$ + $\hat{j}$ + $\sqrt{2}\hat{k}$
& ($\dot{a}$ + $\dot{b}$) is perpendicular to \overrightarrow{c} and projection vector of $\dot{b}$ on $\overrightarrow{a}$ is $\overrightarrow{a}$ then find $\left | \overrightarrow{b} \right |$

maths vectors and transformations introduction to vector algebra algebra of vectors operations on vectors

  1. 6

  2. $\sqrt{22}$

  3. $\sqrt{32}$

  4. 11

Show answer
Correct Option: B 
Let a=i+j+k and c=j-k . If b is a vector satisfying a×b=c and a.b=3, then find b.
maths vectors and transformations introduction to vector algebra algebra of vectors operations on vectors

  1. $\dfrac{1}{3}(5\hat{i}+2\hat{j}+2\hat{k})$

  2. $\dfrac{1}{3}(2\hat{i}+3\hat{j}+\hat{k}$

  3. $\displaystyle 2\overrightarrow{a}$

  4. $\displaystyle -2\overrightarrow{a}$

Show answer
Correct Option: A
Explanation:

Let$b=xi+yj+zk$

Now 
$a.b=x+y+z=3$
Now $a\times b=(i+j+k)(xi+yj+zk)\ \quad=(z-y)i+(x-z)j+(y-x)k=j-k$
Now $z.y=0$ hence $k=y$
Now  $x-z=1$ and $y-x=-1$
Now
$x+y+z=3\1+z+z+z=3\z=\cfrac{2}{3}$
Hence $y=\cfrac{2}{3}\x=\cfrac{5}{3}$
Now $\overrightarrow{b}=\cfrac{(5i+2j+2k)}{3}$

$A$ vector $\vec V$ is inclined at equal angles to axes $OX,OY$ and $OZ$. If $\vec V$ is $6units$, then $\vec V$ is

maths vectors and transformations introduction to vector algebra algebra of vectors operations on vectors

  1. $2\sqrt 3\left( \hat i+\hat j+\hat k right )$

  2. $2\sqrt 3\left( \hat i-\hat j+\hat k right )$

  3. $\sqrt 2\left( \hat i+\hat j+\hat k right )$

  4. $2\sqrt 3\left( \hat i+\hat j-\hat k right )$

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Correct Option: A

$\sum _{ i=1 }^{ n }{ \vec { ai }  } =\vec { 0 } \quad where\quad |\vec { a\quad i\quad | } =1\forall i$ then the value of $\sum _{ 1\le i }^{  }{ \sum _{ <j\le n }^{  }{ \vec { { a } _{ i } }  }  } .\vec { { a } _{ j } } $ is 

maths vectors and transformations introduction to vector algebra algebra of vectors operations on vectors

  1. -n/2

  2. -n

  3. n/2

  4. n

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Correct Option: A