CBSE Questions for Class 12 Commerce Maths Vector Algebra Quiz 9 - MCQExams.com

If vectors $$\vec {A} = 2\hat {i} + 3\hat {j} + 4\hat {k}, \vec {B} = \hat {i} + \hat {j} + 5\hat {k}$$ and $$\vec {C}$$ form a left-handed system, then $$\vec {C}$$ is
  • $$11\hat {i} - 6\hat {j} - \hat {k}$$
  • $$-11\hat {i}+ 6\hat {j} + \hat {k}$$
  • $$11\hat {i} - 6\hat {j} + \hat {k}$$
  • $$-11\hat {i} + 6\hat {j} - \hat {k}$$
If $$2\overrightarrow{a}+3\overrightarrow{b}+4\overrightarrow{c}=0 \Rightarrow \overrightarrow{a}\times \overrightarrow{b} + \overrightarrow{b}\times \overrightarrow{c} +\overrightarrow{c}\times \overrightarrow{a}=$$
  • $$0$$
  • $$3\overrightarrow{a}\times \overrightarrow{b}$$
  • $$3\overrightarrow{b}\times \overrightarrow{c}$$
  • $$3\overrightarrow{c}\times \overrightarrow{a}$$
If $$|\vec{a}|=5, |\vec{a}-\vec{b}|=8$$ and $$|\vec{a}+\vec{b}|=10$$, then $$|\vec{b}|$$ is equal to :
  • $$1$$
  • $$\sqrt{57}$$
  • $$13$$
  • $$14$$
Let $$\overrightarrow{a}=\hat{i}+2\hat{j}+\hat{k},\overrightarrow{b}=\hat{i}-\hat{j}+\hat{k}$$ and $$\overrightarrow{c}=\hat{i}+\hat{j}-\hat{k}$$ . A vector in the plane of $$\overrightarrow{a}$$ and $$\overrightarrow{b}$$, where projection on $$\overrightarrow{c}$$ is $$\dfrac{1}{\sqrt{3}}$$, is
  • $$4\hat{i}-\hat{j}+4\hat{k}$$
  • $$3\hat{i}+\hat{j}-3\hat{k}$$
  • $$2\hat{i}+\hat{j}$$
  • $$4\hat{i}+\hat{j}-4\hat{k}$$
If $$\bar{a}, \bar{b}, \bar{c}$$ are position vectors of the non-collinear points A, B, C respectively, the shortest distance of A and BC is?
  • $$\bar{a}\cdot(\bar{b}-\bar{c})$$
  • $$\bar{b}\cdot(\bar{c}-\bar{a})$$
  • $$|\bar{b}-\bar{a}|$$
  • $$\sqrt{|\vec{b}-\vec{a}|^2-[\cfrac{(\vec{a}-\vec{b}\cdot (\vec{b}-\vec{c})}{|\vec{b}-\vec{c}|}}]^2$$

If $$A,B,C,D$$ be any four points and $$E$$ and $$F$$ be the mid-points of $$AC$$ and $$BD$$, respectively, then $$\vec{AB}+\vec{CB}+\vec{CD}+\vec{AD}$$ is equal to
  • $$3\vec{EF}$$
  • $$4\vec{EF}$$
  • $$4\vec{FE}$$
  • $$3\vec{FE}$$
Let $$\overrightarrow{a}=2\hat{i}-3\hat{j}+6\hat{k}$$ and $$\overrightarrow{b}=-2\hat{i}+3\hat{j}-\hat{k}$$ then projection of $$\overrightarrow{a}$$ on $$\overrightarrow{b} :$$ projection of $$\overrightarrow{b}$$ on $$\overrightarrow{a}=$$
  • $$3:7$$
  • $$7:3$$
  • $$-4$$
  • $$3$$
Let $$\overrightarrow{u}=\hat{i}+\hat{j},\overrightarrow{v}=\hat{i}-\hat{j},\overrightarrow{w}=\hat{i}+2\hat{j}+3\hat{k}$$ If $$\hat{n}$$ is a unit vector such that $$\overrightarrow{u}.\hat{n}=0$$ and $$\overrightarrow {v}.\hat{n}=0$$, then $$\left|\overrightarrow {w}.\hat{n}\right|=$$
  • $$0$$
  • $$1$$
  • $$2$$
  • $$3$$
Let $$\overrightarrow{a}=\hat{i}+2\hat{j}+\hat{k},\overrightarrow{b}=\hat{i}-\hat{j}+\hat{k}$$ and $$\overrightarrow{c}=\hat{i}+\hat{j}-\hat{k}$$ . A vector in the plane of $$\overrightarrow{a}$$ and $$\overrightarrow{b}$$, where projection on $$\overrightarrow{c}$$ is $$\dfrac{1}{\sqrt{3}}$$, is
  • $$4\hat{i}-\hat{j}+4\hat{k}$$
  • $$3\hat{i}+\hat{j}-3\hat{k}$$
  • $$2\hat{i}+\hat{j}$$
  • $$4\hat{i}+\hat{j}-4\hat{k}$$
let $$\overrightarrow{u},\overrightarrow{v},\overrightarrow{w}$$ be such that $$\left|\overrightarrow{u}\right|=1,\left|\overrightarrow{v}\right|=2,\left|\overrightarrow{w}\right|=3$$. If the projection of $$\overrightarrow{v}$$ along $$\overrightarrow{u}$$ is equal to the projection of $$\overrightarrow{w}$$ along $$\overrightarrow{u}$$ and $$\overrightarrow{v},\overrightarrow{w}$$ are perpendicular to each other, then$$\left|\overrightarrow{u}-\overrightarrow{v}+\overrightarrow{w}\right|=$$
  • $$2$$
  • $$\sqrt{17} $$
  • $$\sqrt{14}$$
  • $$\sqrt{15}$$
If $$\vec{a}+\vec{b}+\vec{c}=vec{0}$$ then $$\vec{a}\times \vec{b}=?$$
  • $$\vec{c}\times \vec{b}$$
  • $$\vec{b}\times \vec{c}$$
  • $$\vec{a}\times \vec{c}$$
  • $$2\vec{b}\times \vec{c}$$
The ratio in which $$i+2j+3k$$ divides the join of $$-2i+3j+5k$$ and $$7i-k$$ is?
  • $$-3:2$$
  • $$1:2$$
  • $$2:3$$
  • $$-4:3$$
If $$\overrightarrow{a}\times \overrightarrow{b}=\overrightarrow{c}$$ and $$\overrightarrow{b}\times \overrightarrow{c}=\overrightarrow{a}$$, then 
  • $$\left|\overrightarrow{a}\right|=\left|\overrightarrow{b}\right|=\left|\overrightarrow{c}\right| $$
  • $$\left|\overrightarrow{a}\right|=\left|\overrightarrow{c}\right|,\left|\overrightarrow{b}\right|=1$$
  • $$\overrightarrow{c}\times \overrightarrow{a}=\overrightarrow{b}$$
  • $$\overrightarrow{c}\times \overrightarrow{a}=\left[\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}\right]\overrightarrow{b}$$
Let $$\left| \overline { a } +\overline { b }  \right| =\left| \overline { a } -\overline { b }  \right| $$. If $$\left| \overline { a } \times \overline { b }  \right| =\lambda \left| \overline { a }  \right| $$, then $$\lambda=$$
  • $$\left| \overline { a } \right| $$
  • $$\left| \overline { b } \right| $$
  • $$1$$
  • $$2$$
The projection of the vector $$\hat{i}-2\hat{j}+\hat{k}$$ on he vector $$4\hat{i}-4\hat{j}+7\hat{k}$$ is equal to:
  • $$\dfrac{19}{9}$$
  • $$\dfrac{9}{19}$$
  • $$\dfrac{\sqrt{3}}{19}$$
  • $$\dfrac{19}{\sqrt{3}}$$
If $$\left|\overrightarrow{a} \right|=1$$, the projection of $$\overrightarrow{r}$$ along $$\overrightarrow{a} $$ is $$2$$ and $$\overrightarrow{a}\times \overrightarrow{r}+\overrightarrow{b}=\overrightarrow{r}$$, then $$\overrightarrow{r}=$$
  • $$\dfrac{1}{2}\left[\overrightarrow{a}-\overrightarrow{b}+\overrightarrow{a}\times \overrightarrow{b}\right]$$
  • $$\dfrac{1}{2}\left[2\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{a}\times \overrightarrow{b}\right]$$
  • $$\overrightarrow{a}+\overrightarrow{a}\times \overrightarrow{b}$$
  • $$\overrightarrow{a}-\overrightarrow{a}\times \overrightarrow{b}$$
Let $$A,B,C$$ be distinct point with position vectors $$\hat{i}+\hat{j}$$, $$\hat{i}-\hat{j}$$, $$p\hat{i}-q\hat{j}+r\hat{k}$$ respectively. Points $$A,B,C$$ are collinear, then which of the following can be correct:
  • $$p=q=r=1$$
  • $$p=q=r=0$$
  • $$p=q=2,r=0$$
  • $$p=1,q=2,r=0$$
$$D,E\ and \ F$$ are the mid-point of the sides $$BC,CA \ and \ AB$$ respectively of the triangle $$ABC$$. Which of the following is true?
  • $${ \xrightarrow { AB= } \xrightarrow { 2ED } }$$
  • $${ \xrightarrow { AB= } \xrightarrow { 2DE } }$$
  • $${ \xrightarrow { AB= } \xrightarrow { ED } }$$
  • $${ \xrightarrow { AB= } \xrightarrow { 2DF } }$$
If $$\overrightarrow{a}\times \overrightarrow{b}=\overrightarrow{b}\times \overrightarrow{c}=\overrightarrow{c}\times \overrightarrow{a}\neq 0$$ then $$\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=$$
  • $$-\overrightarrow{b}$$
  • $$2\overrightarrow{a} $$
  • $$0$$
  • none of these

A unit vector perpendicular to the plane of the triangle ABC with the position vectors  $$\vec a\,\,\,\,\vec b\,\,\vec c$$ of the vectors A,B,C, is 
  • $$\frac{{(\vec a\,\, \times \,\,\vec b + \,\vec b\,\, \times \,\,\vec c\,\, + \vec c \times \vec a)\,\,\,}}{\Delta }$$
  • $$\frac{{(\vec a\,\, \times \,\,\vec b + \,\vec b\,\, \times \,\,\vec c\,\, + \vec c \times \vec a)\,\,\,}}{{2\Delta }}$$
  • $$\frac{{(\vec a\,\, \times \,\,\vec b + \,\vec b\,\, \times \,\,\vec c\,\, + \vec c \times \vec a)\,\,\,}}{{4\Delta }}$$
  • none of these
If $$\overrightarrow { a } =2\hat { i } +\hat { j } +\hat { k } ,\overrightarrow { b } =3\hat { i } -4\hat { j } +2\hat { k } ,\overrightarrow { c } =\hat { i } -2\hat { j } +2\hat { k } $$ then the projection of $$\overrightarrow { a } +\overrightarrow { b } $$ on $$\overrightarrow { c } $$ is
  • $$\cfrac{17}{3}$$
  • $$\cfrac{5}{3}$$
  • $$\cfrac{4}{3}$$
  • $$\cfrac{17}{\sqrt{43}}$$
If $$|a|=5.|\vec{b}|=4$$, and $$|c|=3$$. then what will be the value of $$\vec{a}.\vec{b}+\vec{b}.\vec{c}+\vec{c}.\vec{a}$$ given that $$\vec{a}+\vec{b}+\vec{c}=0$$
  • $$25$$
  • $$50$$
  • $$-25$$
  • $$-50$$
If $$A(6,3,2),B(5,1,4),C(3,-4,7), D(0,2,5)$$ are four points, then projection of $$CD$$ on $$AB$$ is
  • $$-\cfrac{13}{3}$$
  • $$-\cfrac{13}{7}$$
  • $$-\cfrac{3}{13}$$
  • $$-\cfrac{7}{13}$$
If $$\vec a,\vec b$$ and $${\vec c}$$ are unit vectors, then $${\left| {\vec a + \vec b} \right|^2} + {\left| {\vec b - \vec c} \right|^2} + {\left| {\vec c - \vec a} \right|^2}$$ does NOT exceed 
  • 4
  • 9
  • 8
  • 6
If $$\overline { a } $$ and $$\overline { b } $$ include an angle of $${120}^{o}$$ and their magnitudes are $$2$$ and $$\sqrt{3}$$ then $$\overline { a } .\overline { b } $$ is
  • $$3$$
  • $$-\sqrt{3}$$
  • $$\sqrt{3}$$
  • $$-3$$
If $$\left| {\vec a} \right| = 2,\left| {\vec b} \right| = 3$$ and  $$\left| {2\vec a - \vec b} \right| = 5,$$ then  $$\left| {2\vec a + \vec b} \right|$$ equals:
  • $$17$$
  • $$7$$
  • $$5$$
  • $$1$$
If the vectors $$\overrightarrow { a } =\hat { i } -\hat { j } +2\hat { k } ;\overrightarrow { b } =2\hat { i } +4\hat { j } +\hat { k } ;\overrightarrow { c } =\lambda \hat { i } +\hat { j } +\mu \hat { k } $$ are mutually orthogonal, then $$(\lambda,\mu)=$$
  • $$(2,-3)$$
  • $$(-2,3)$$
  • $$(3,-2)$$
  • $$(-3,2)$$
If $$\left| \overrightarrow { a }  \right| =3,\left| \overrightarrow { b }  \right| =4$$, if $$\left( \overrightarrow { a } +\lambda \overrightarrow { b }  \right) $$ is perpendicular to $$\left( \overrightarrow { a } -\lambda \overrightarrow { b }  \right) $$ then $$\lambda =$$
  • $$\cfrac{9}{16}$$
  • $$\cfrac{3}{5}$$
  • $$\cfrac{3}{4}$$
  • $$\cfrac{4}{3}$$
Let $$\bar{a},\bar{b}$$ be two noncollinear vectors. If $$A=(x+4y)\bar{a}+(2x+y+1)\bar{b},$$
$$B=(y-2x+2)\bar{a}+(2x-3y-1)\bar{b} \quad and \quad 3A=2B$$ then (x,y) =
  • (1,2)
  • (1,-2)
  • (2,-1)
  • (-2,-1)
The projection of the vector $$2\hat i + \hat j - 3\hat k$$ on  the vector $$\hat i - 2\hat j - \hat k$$
  • $$ - \dfrac{3}{{\sqrt {14} }}$$
  • $$ \dfrac{3}{{\sqrt {14} }}$$
  • $$ - \sqrt {\dfrac{3}{2}} $$
  • $$ \sqrt {\dfrac{3}{2}} $$
The position vectors of the points A,B,C are $$\overline { i } + 2 \overline { j } - \overline { k } , \overline { i } + \overline { j } + \overline { k } , 2 \overline { i } + 3 \overline { j } + 2 \overline { k }$$ respectively. If A is chosen as the origin then the position vectors of B and C are 
  • $$\overline { i } + 2 \overline { k } , \overline { i } + \overline { j } + 3 \overline { k }$$
  • $$\overline { j } + 2 \overline { k } , \vec { i } + \overline { j } + 3 \overline { k }$$
  • $$- \overline { j } + 2 \overline { k } , \vec { i } - \overline { j } + 3 \overline { k }$$
  • $$- \overline { j } + 2 \overline { k } , \overline { i } + \overline { j } + 3 \overline { k }$$
Given $$\vec{\alpha} = 3\hat{i} + \hat{j} + 2\hat{k}\ ,\ \vec{\beta} = \hat{i} - 2\hat{j} - 4\hat{k}$$ are the position vectors of the points $$A$$ and $$B$$. Then the distance of the point $$-\hat{i} + \hat{j} + \hat{k}$$ from the passing through $$B$$ and perpendicular to $$AB$$ is 
  • $$-7$$
  • $$10$$
  • $$15$$
  • $$20$$
If $$M$$ and $$N$$ are the mid-points of the diagonals $$AC$$ and $$BD$$ respectively of a quadrilateral $$ABCD$$, then the value of $$\overline { AB } +\overline { AD } +\overline { CB } +\overline { CD } $$
  • $$2\overline { MN } $$
  • $$2\overline { NM } $$
  • $$4\overline { NM } $$
  • $$4\overline { MN } $$
If $$S$$ is the circumcentre, $$O$$ is the orthocentre of $$\triangle{ABC}$$, then $$\overline { SA } +\overline { SB } +\overline { SB } $$ equals
  • $$\overline { SO } $$
  • $$2\overline { SO } $$
  • $$\overline { OS } $$
  • $$2\overline { OS } $$
A (1,-1,-1) , B (2,1,-2) and C (-5,2,-6) are the position vectors of the vertices of triangle ABC  The length of the bisector of its internal angle at A is:
  • $$\dfrac{\sqrt{10}}{4}$$
  • $$\dfrac{3\sqrt{10}}{4}$$
  • $$\sqrt{10}$$
  • none
If $$\overline { a } =\left( 2\overline { i } -10\overline { j } +6\overline { k }  \right) ;\overline { b } =\left( 5\overline { i } -3\overline { j } +\overline { k }  \right) $$. The ratio of projection of $$\overline { a } $$ on $$\overline { b } $$ to projection of $$\overline { b } $$ on $$\overline { a } $$ is
  • $$2:1$$
  • $$1:2$$
  • $$2:3$$
  • $$3:2$$
Let $$\vec a$$ and $$\vec b$$ be two unit vectors such that $$\left| {\vec a + \vec b} \right| = \sqrt 3$$. If $$\vec c = \vec a + 2\vec b + 3(\vec a \times \vec b)$$, then $$2\left| {\vec c} \right|$$ is equal to:
  • $$\sqrt {55} $$
  • $$\sqrt {51} $$
  • $$\sqrt {43} $$
  • $$\sqrt {37} $$
The $$X$$ & $$Y$$ components of vector A have numerical values $$6$$ each & that of $$(A+B)$$ have numerical values $$10$$ and $$9$$ What is the numerical value of $$B$$ ?
  • $$2$$
  • $$3$$
  • $$4$$
  • $$5$$
$$ABC$$ is an isosceles triangle right angled at $$A$$. Force of magnitude $$2\sqrt{2},5$$ and $$6$$ act along $$\overline { BC } ,\overline { CA } ,\overline { AB } $$ respectively. The magnitude of their resultant force is
  • $$4$$
  • $$5$$
  • $$11+2\sqrt{2}$$
  • $$30$$
If $$\overline { a } $$ is a vector of magnitude $$\sqrt{3}$$ and $$\overline { b } $$ is unit vector making an angle $$\tan ^{ -1 }{ \left( 1/\sqrt { 2 }  \right)  } $$ with $$\overline { a } $$ then projection of $$\overline { a } $$ on $$\overline { b } $$ is
  • $$\cfrac{\sqrt{3}}{2}$$
  • $$\sqrt{2}$$
  • $$\sqrt{3}$$
  • $$\sqrt{6}$$
In the figure given below $$\bar { AE },$$ $$\bot\bar { DB },$$ $$\bar { CF }$$ $$\bot\bar { BD },$$ $$\bot\bar { DF }$$ $$=$$ $$\bar { BE },$$ $$\bar { AD },$$ $$=$$ $$\bar { BC }$$  
then $$\bar { DC }=\bar { AB }.$$  
1211158_35bf6200029b48098b260ed669c240bd.png
  • True
  • False
Given a parallelogram $$ABCD$$. If $$|\vec{AB}=a, |\vec{AD}|=b$$ and $$|vec{AC}|=c$$, then $$|\vec{DB}|.|\vec{AB}|$$ has the
  • $$\dfrac{3a^{2}+b^{2}-c^{2}}{2}$$
  • $$\dfrac{a^{2}+3b^{2}-c^{2}}{2}$$
  • $$\dfrac{a^{2}+b^{2}-3c^{2}}{2}$$
  • $$none$$
If $$|\bar {a}-\bar {b}|=|\bar {a}|=|\bar {b}|$$, where $$\bar a$$ and $$\bar b$$ are non zero vecrors then the angle between $$\bar {a}-\bar {b}$$ and $$\bar b$$ is
  • $$120^{o}$$
  • $$45^{o}$$
  • $$60^{o}$$
  • $$90^{o}$$
The vector $$T + 2 \overline { y } + 2 k$$ restated through an angle $$\theta$$ and doubled in magnitude then it becomes $$2 T + ( 2 x + 2 ) \} + ( 6 x - 2 ) k$$ values of $$x$$ are 
  • $$1,\frac { 1 } { 3 }$$
  • $$- 1 , \frac { 1 } { 3 } =$$
  • $$1,\frac { - 1 } { 3 }$$
  •  $$( 0,3 )$$
If AD, BE and CF are $$\Delta ABC$$, then $$\\ \vec { AD } +\vec { BE } \vec { +CF } $$
  • $$\vec { 0 } $$
  • 1
  • 0
  • 2
Let $$\overline a,\overline b, \overline c$$ be three vectors such that $$\overline a \ne 0$$, and $$\overline a \times \overline b= 2\overline a \times \overline c, |\overline a|=|\overline c|=1, |\overline a|$$
$$|\overline b \times \overline c|= \sqrt 15$$. If $$\overline b - 2 \overline c= \lambda \overline a$$, then $$\lambda$$ equals to
  • $$1$$
  • $$ ± 4$$
  • $$3$$
  • $$-2$$
 $$\bar{a}$$, $$\bar{b}$$ are unit vectors such that $$\mid\bar{a}\times\bar{b}\mid = \bar{a}. \bar{b}$$ , then $$\mid\bar{a}+\bar{b}\mid^2 =$$
  • $$2$$
  • $$2+\sqrt{2}$$
  • $$2-\sqrt{2}$$
  • $$\sqrt{2}$$
The length of the projection of the line segment joining the points $$(5, -1, 4)$$ and $$(4, -1, 3)$$ on the plane , $$ x+ y+ z = 7$$ is : 
  • $$\dfrac{2}{3}$$
  • $$\dfrac{1}{3}$$
  • $$\sqrt{\dfrac{2}{3}}$$
  • $$\dfrac{2}{\sqrt{3}}$$
$$\vec{a},\vec{b},\vec{c},\vec{d}$$ are the position vectors of four coplanar points A,B,C,D respectively. If no three of them are collinear and $$|\vec{a}-\vec{d}|=|\vec{b}-\vec{d}|=|\vec{c}-\vec{d}|$$ then for triangle ABC, D is
  • centroid
  • orthocenter
  • incenter
  • circumcenter
If the position vectors of P, Q are respectively 5a + 4b and 3a - 2b then $$\vec {QP}$$ =
  • $$2a + 6b$$
  • $$2a  -  6b$$
  • $$2a  +  5b$$
  • $$2a  -  5b$$
0:0:1


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