JEE Main & Advanced Mathematics Vector Algebra Question Bank

done Critical Thinking Total Questions - 40

• question_answer1)
  Three forces of magnitudes 1, 2, 3 dynes meet in a point and act along diagonals of three adjacent faces of a cube. The resultant force is [MNR 1987]
  A)
  114 dyne done clear
  B)
  6 dyne done clear
  C)
  5 dyne done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer2)
  The vectors b and c are in the direction of north-east and north-west respectively and |b|=|c|= 4. The magnitude and direction of the vector d = c - b, are [Roorkee 2000]
  A)
  \[4\sqrt{2},\] towards north done clear
  B)
  \[4\sqrt{2}\], towards west done clear
  C)
  4, towards east done clear
  D)
  4, towards south done clear
  View Solution play_arrow

• question_answer3)
  If a, b and c are unit vectors, then \[|\mathbf{a}-\mathbf{b}{{|}^{2}}+|\mathbf{b}-\mathbf{c}{{|}^{2}}+|\mathbf{c}-\mathbf{a}{{|}^{2}}\]  does not exceed [IIT Screening 2001]
  A)
  4 done clear
  B)
  9 done clear
  C)
  8 done clear
  D)
  6 done clear
  View Solution play_arrow

• question_answer4)
  The vectors \[\overrightarrow{AB}=3\mathbf{i}+5\mathbf{j}+4\mathbf{k}\] and \[\overrightarrow{AC}=5\mathbf{i}-5\mathbf{j}+2\mathbf{k}\] are the sides of a triangle ABC. The length of the median through A is [UPSEAT 2004]
  A)
  \[\sqrt{13}\] unit done clear
  B)
  \[\theta \] unit done clear
  C)
  5 unit done clear
  D)
  10 unit done clear
  View Solution play_arrow

• question_answer5)
  Let the value of \[\mathbf{p}=(x+4y)\,\mathbf{a}+(2x+y+1)\,\mathbf{b}\] and \[\mathbf{q}=(y-2x+2)\,\mathbf{a}+(2x-3y-1)\,\mathbf{b},\] where a and b are non-collinear vectors. If \[3\mathbf{p}=2\mathbf{q},\] then the value of x and y will be [RPET 1984; MNR 1984]
  A)
  - 1, 2 done clear
  B)
  2, - 1 done clear
  C)
  1, 2 done clear
  D)
  2, 1 done clear
  View Solution play_arrow

• question_answer6)
  The points D, E, F divide BC, CA and AB of the triangle ABC in the ratio 1 : 4, 3 : 2 and 3 : 7 respectively and the point K divides AB in the ratio \[1:3\], then \[(\overrightarrow{AD}+\overrightarrow{BE}+\overrightarrow{CF})\,\,:\,\,\overrightarrow{CK}\] is equal to [MNR 1987]
  A)
  1 : 1                                           done clear
  B)
  2 : 5 done clear
  C)
  5 : 2                                           done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer7)
  If two vertices of a triangle are \[\mathbf{i}-\mathbf{j}\] and \[\mathbf{j}+\mathbf{k}\], then the third vertex can be [Roorkee 1995]
  A)
  \[\mathbf{i}+\mathbf{k}\] done clear
  B)
  \[\mathbf{i}-2\mathbf{j}-\mathbf{k}\] done clear
  C)
  \[\mathbf{i}-\mathbf{k}\] done clear
  D)
  \[2\mathbf{i}-\mathbf{j}\] done clear
  E)
  All the above done clear
  View Solution play_arrow

• question_answer8)
  If a of magnitude 50 is collinear with the vector \[\mathbf{b}=6\,\mathbf{i}-8\,\mathbf{j}-\frac{15\,\mathbf{k}}{2},\] and makes an acute angle with the positive direction of z-axis, then the vector a is equal to [Pb. CET 2004]
  A)
  \[24\,\mathbf{i}-32\,\mathbf{j}+30\,\mathbf{k}\] done clear
  B)
  \[-24\,\mathbf{i}+32\,\mathbf{j}+30\,\mathbf{k}\] done clear
  C)
  \[16\,\mathbf{i}-16\,\mathbf{j}-15\,\mathbf{k}\] done clear
  D)
  \[-12\,\mathbf{i}+16\,\mathbf{j}-30\,\mathbf{k}\] done clear
  View Solution play_arrow

• question_answer9)
  If three non-zero vectors are \[\mathbf{a}={{a}_{1}}\mathbf{i}+{{a}_{2}}\mathbf{j}+{{a}_{3}}\mathbf{k},\] \[\mathbf{b}={{b}_{1}}\mathbf{i}+{{b}_{2}}\mathbf{j}+{{b}_{3}}\mathbf{k}\] and \[\mathbf{c}={{c}_{1}}\mathbf{i}+{{c}_{2}}\mathbf{j}+{{c}_{3}}\mathbf{k}.\] If c is the unit vector perpendicular to the vectors a and b and the angle between a and b is \[\frac{\pi }{6},\] then \[{{\left| \,\begin{matrix}  {{a}_{1}} & {{a}_{2}} & {{a}_{3}}  \\  {{b}_{1}} & {{b}_{2}} & {{b}_{3}}  \\    {{c}_{1}} & {{c}_{2}} & {{c}_{3}}  \\ \end{matrix}\, \right|}^{2}}\] is equal to    [IIT 1986]
  A)
  0 done clear
  B)
  \[\frac{3\,(\Sigma a_{1}^{2})\,(\Sigma b_{1}^{2})\,(\Sigma c_{1}^{2})}{4}\] done clear
  C)
  1     done clear
  D)
  \[\frac{(\Sigma a_{1}^{2})\,(\Sigma b_{1}^{2})}{4}\] done clear
  View Solution play_arrow

• question_answer10)
  Let the unit vectors a and b be perpendicular and the unit vector c be inclined at an angle q to both a and b. If \[\mathbf{c}=\alpha \,\mathbf{a}+\beta \,\mathbf{b}+\gamma \,(\mathbf{a}\times \mathbf{b}),\] then [Orissa JEE 2003]
  A)
  \[\alpha =\beta =\cos \theta ,\,\,{{\gamma }^{2}}=\cos \,\,2\theta \] done clear
  B)
  \[\alpha =\beta =\cos \theta ,\,\,{{\gamma }^{2}}=-\cos \,\,2\theta \] done clear
  C)
  \[\alpha =\cos \theta ,\,\,\beta =\sin \theta ,\,\,{{\gamma }^{2}}=\cos \,\,2\theta \] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer11)
  The vector \[\mathbf{a}+\mathbf{b}\] bisects the angle between the vectors a and b, if
  A)
  \[|\mathbf{a}|\,=\,|\mathbf{b}|\] done clear
  B)
  \[|\mathbf{a}|\,=\,|\mathbf{b}|\] or angle between a and b is zero done clear
  C)
  \[|\mathbf{a}|\,\,=m\,|\mathbf{b}|\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer12)
  The points \[O,\,A,\,B,\,C,\,D\] are such that \[\overrightarrow{OA}=\mathbf{a},\] \[\overrightarrow{OB}=\mathbf{b},\,\] \[\overrightarrow{OC}=2\mathbf{a}+3\mathbf{b}\] and \[\overrightarrow{OD}=\mathbf{a}-2\mathbf{b}.\] If \[|\mathbf{a}|\,=3\,|\mathbf{b}|,\] then the angle between \[\overrightarrow{BD}\] and \[\overrightarrow{AC}\] is
  A)
  \[\frac{\pi }{3}\] done clear
  B)
  \[\frac{\pi }{4}\] done clear
  C)
  \[\frac{\pi }{6}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer13)
  If \[\overrightarrow{A}=\mathbf{i}+2\mathbf{j}+3\mathbf{k},\,\,\,\overrightarrow{B}=-\mathbf{i}+2\mathbf{j}+\mathbf{k}\] and \[\overrightarrow{C}=3\mathbf{i}+\mathbf{j},\] then the value of t such that \[\overrightarrow{A}+t\overrightarrow{B}\] is at right angle to vector \[3\mathbf{i}+4\mathbf{j}\] is       [RPET 2002]
  A)
  2 done clear
  B)
  4 done clear
  C)
  5 done clear
  D)
  6 done clear
  View Solution play_arrow

• question_answer14)
  Let \[\mathbf{b}=4\mathbf{i}+3\mathbf{j}\] and c be two vectors perpendicular to each other in the xy-plane. All vectors in the same plane having projections 1 and 2 along b and c respectively, are given by [IIT 1987]
  A)
  \[2\mathbf{i}-\mathbf{j},\,\,\frac{2}{5}\mathbf{i}+\frac{11}{5}\mathbf{j}\] done clear
  B)
  \[2\mathbf{i}+\mathbf{j},\,\,-\frac{2}{5}\mathbf{i}+\frac{11}{5}\mathbf{j}\] done clear
  C)
  \[2\mathbf{i}+\mathbf{j},\,-\frac{2}{5}\mathbf{i}-\frac{11}{5}\mathbf{j}\] done clear
  D)
  \[2\mathbf{i}-\mathbf{j},\,\,-\frac{2}{5}\mathbf{i}+\frac{11}{5}\mathbf{j}\] done clear
  View Solution play_arrow

• question_answer15)
  Let \[\mathbf{a}=2\mathbf{i}-\mathbf{j}+\mathbf{k},\,\,\mathbf{b}=\mathbf{i}+2\mathbf{j}-\mathbf{k}\] and \[\mathbf{c}=\mathbf{i}+\mathbf{j}-2\mathbf{k}\] be three vectors. A vector in the plane of b and c whose projection on a is of magnitude \[\sqrt{2/3}\] is [IIT 1993; Pb. CET 2004]
  A)
  \[2\mathbf{i}+3\mathbf{j}-3\mathbf{k}\] done clear
  B)
  \[2\mathbf{i}+3\mathbf{j}+3\mathbf{k}\] done clear
  C)
  \[-\,2\mathbf{i}-\mathbf{j}+5\mathbf{k}\] done clear
  D)
  \[2\mathbf{i}+\mathbf{j}+5\mathbf{k}\] done clear
  View Solution play_arrow

• question_answer16)
  A vector a has components 2p and 1 with respect to a rectangular cartesian system. The system is rotated through a certain angle about the origin in the anti-clockwise sense. If a has components p+1 and 1 with respect to the new system, then [IIT 1984]
  A)
  \[p=0\] done clear
  B)
  \[p=1\] or \[-\frac{1}{3}\] done clear
  C)
  \[p=-1\] or \[\frac{1}{3}\] done clear
  D)
  \[p=1\] or \[-1\] done clear
  View Solution play_arrow

• question_answer17)
  If \[\mathbf{u}=2\,\mathbf{i}+2\mathbf{j}-\mathbf{k}\]and \[\mathbf{v}=6\,\mathbf{i}-3\,\mathbf{j}+2\,\mathbf{k},\] then a unit vector perpendicular to both u and v is [MP PET 1987]
  A)
  \[\mathbf{i}-10\mathbf{j}-18\mathbf{k}\] done clear
  B)
  \[\frac{1}{\sqrt{17}}\,\left( \frac{1}{5}\mathbf{i}-2\mathbf{j}-\frac{18}{5}\mathbf{k} \right)\] done clear
  C)
  \[\frac{1}{\sqrt{473}}\,(7\mathbf{i}-10\mathbf{j}-18\mathbf{k})\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer18)
  If \[\mathbf{a}=2\mathbf{i}+\mathbf{k},\,\,\mathbf{b}=\mathbf{i}+\mathbf{j}+\mathbf{k}\] and \[\mathbf{c}=4\mathbf{i}-3\mathbf{j}+7\mathbf{k}.\] If \[\mathbf{d}\times \mathbf{b}=\mathbf{c}\times \mathbf{b}\] and \[\mathbf{d}\,.\,\mathbf{a}=0,\] then d will be [IIT 1990]
  A)
  \[\mathbf{i}+8\mathbf{j}+2\mathbf{k}\] done clear
  B)
  \[\mathbf{i}-8\mathbf{j}+2\mathbf{k}\] done clear
  C)
  \[-\mathbf{i}+8\mathbf{j}-\mathbf{k}\] done clear
  D)
  \[-\mathbf{i}-8\mathbf{j}+2\mathbf{k}\] done clear
  View Solution play_arrow

• question_answer19)
  If \[\mathbf{a}\times \mathbf{r}=\mathbf{b}+\lambda \mathbf{a}\] and \[\mathbf{a}\,\,.\,\,\mathbf{r}=3,\] where \[\mathbf{a}=2\mathbf{i}+\mathbf{j}-\mathbf{k}\] and \[\mathbf{b}=-\mathbf{i}-2\mathbf{j}+\mathbf{k},\] then r and l are equal to 
  A)
  \[\mathbf{r}=\frac{7}{6}\mathbf{i}+\frac{2}{3}\mathbf{j},\,\,\lambda =\frac{6}{5}\] done clear
  B)
  \[\mathbf{r}=\frac{7}{6}\mathbf{i}+\frac{2}{3}\mathbf{j},\,\,\lambda =\frac{5}{6}\] done clear
  C)
  \[\mathbf{r}=\frac{6}{7}\mathbf{i}+\frac{2}{3}\mathbf{j},\,\,\lambda =\frac{6}{5}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer20)
  Let the vectors a, b, c and d be such that\[(\mathbf{a}\times \mathbf{b})\times (\mathbf{c}\times \mathbf{d})=0\]. Let \[{{P}_{1}}\] and \[{{P}_{2}}\] be planes determined by pair of vectors a, b and c, d respectively. Then the angle between \[{{P}_{1}}\] and \[{{P}_{2}}\] is  [IIT Screening 2000; MP PET 2004] 
  A)
  \[{{0}^{o}}\] done clear
  B)
  \[\frac{\pi }{4}\] done clear
  C)
  \[\frac{\pi }{3}\] done clear
  D)
  \[\frac{\pi }{2}\] done clear
  View Solution play_arrow

• question_answer21)
  If \[\mathbf{a}=\mathbf{i}+\mathbf{j}+\mathbf{k},\,\,\mathbf{a}\,.\,\mathbf{b}=1\] and \[\mathbf{a}\times \mathbf{b}=\mathbf{j}-\mathbf{k},\] then \[\mathbf{b}=\] [IIT Screening 2004]
  A)
  \[\mathbf{i}\] done clear
  B)
  \[\mathbf{i}-\mathbf{j}+\mathbf{k}\] done clear
  C)
  \[2\mathbf{j}-\mathbf{k}\] done clear
  D)
  \[2\mathbf{i}\] done clear
  View Solution play_arrow

• question_answer22)
  The position vectors of the vertices of a quadrilateral ABCD are \[a,\,b,\,c\] and d respectively. Area of the quadrilateral formed by joining the middle points of its sides is [Roorkee 2000]
  A)
  \[\frac{1}{4}\,|a\times b+b\times d+d\times a|\] done clear
  B)
  \[\frac{1}{4}\,\left| b\times  c+c\times d+a\times d+b\times a \right|\] done clear
  C)
  \[\frac{1}{4}\,\left| a\times  b+b\times c+c\times d+d\times a \right|\] done clear
  D)
  \[\frac{\text{1}}{\text{4}}\text{ }\!\!|\!\!\text{ b }\!\!\times\!\!\text{ c+c }\!\!\times\!\!\text{ d+d }\!\!\times\!\!\text{ b }\!\!|\!\!\text{ }\]. done clear
  View Solution play_arrow

• question_answer23)
  The moment about the point \[M(-2,\,4,\,-6)\] of the force represented in magnitude and position by \[\overrightarrow{AB}\] where the points A and B have the co-ordinates \[(1,\,2,\,-3)\] and \[(3,\,-4,\,2)\] respectively, is [MP PET 2000]
  A)
  8i - 9j - 14k done clear
  B)
  2i - 6j + 5k done clear
  C)
  - 3i + 2j - 3k done clear
  D)
  - 5i + 8j - 8k done clear
  View Solution play_arrow

• question_answer24)
  If the vectors \[a\mathbf{i}+\mathbf{j}+\mathbf{k},\,\,\mathbf{i}+b\mathbf{j}+\mathbf{k}\] and \[\mathbf{i}+\mathbf{j}+c\mathbf{k}\] \[(a\ne b\ne c\ne 1)\] are coplanar, then the value of \[\frac{1}{1-a}+\frac{1}{1-b}+\frac{1}{1-c}=\] [BIT Ranchi 1988; RPET 1987; IIT 1987; DCE 2001; MP PET 2004; Orissa JEE 2005]
  A)
  - 1 done clear
  B)
  \[-\frac{1}{2}\] done clear
  C)
  \[\frac{1}{2}\] done clear
  D)
  1 done clear
  View Solution play_arrow

• question_answer25)
  If \[\alpha \,(\mathbf{a}\times \mathbf{b})+\beta \,(\mathbf{b}\times \mathbf{c})+\gamma \,(\mathbf{c}\times \mathbf{a})=\mathbf{0}\] and at least one of the numbers \[\alpha ,\,\,\beta \] and \[\gamma \] is non-zero, then the vectors a, b and c are                                  
  A)
  Perpendicular done clear
  B)
  Parallel done clear
  C)
  Coplanar done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer26)
  The volume of the tetrahedron, whose vertices are given by the vectors \[-\mathbf{i}+\mathbf{j}+\mathbf{k},\,\,\mathbf{i}-\mathbf{j}+\mathbf{k}\] and \[\mathbf{i}+\mathbf{j}-\mathbf{k}\]with reference to the fourth vertex as origin, is
  A)
  \[\frac{5}{3}\]cubic unit done clear
  B)
  \[\frac{2}{3}\] cubic unit done clear
  C)
  \[\frac{3}{5}\]cubic unit done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer27)
  Let \[\mathbf{a}=\mathbf{i}-\mathbf{j},\,\,\mathbf{b}=\mathbf{j}-\mathbf{k},\,\,\mathbf{c}=\mathbf{k}-\mathbf{i}.\] If \[\mathbf{\hat{d}}\] is a unit vector such that \[\mathbf{a}\,.\,\mathbf{\hat{d}}=0=[\mathbf{b}\,\,\mathbf{c}\,\,\mathbf{\hat{d}}],\] then \[\mathbf{\hat{d}}\] is equal to [IIT 1995]
  A)
  \[\pm \frac{\mathbf{i}+\mathbf{j}-\mathbf{k}}{\sqrt{3}}\] done clear
  B)
  \[\pm \frac{\mathbf{i}+\mathbf{j}+\mathbf{k}}{\sqrt{3}}\] done clear
  C)
  \[\pm \frac{\mathbf{i}+\mathbf{j}-2\mathbf{k}}{\sqrt{6}}\] done clear
  D)
  \[\pm \,\,\mathbf{k}\] done clear
  View Solution play_arrow

• question_answer28)
  The value of 'a' so that the volume of parallelopiped formed by \[\mathbf{i}+a\mathbf{j}+\mathbf{k},\mathbf{j}+a\,\mathbf{k}\] and \[a\,\mathbf{i}+\mathbf{k}\] becomes minimum is [IIT Screening 2003]
  A)
  - 3 done clear
  B)
  3 done clear
  C)
  \[\frac{1}{\sqrt{3}}\] done clear
  D)
  \[\sqrt{3}\] done clear
  View Solution play_arrow

• question_answer29)
  If b and c are any two non-collinear unit vectors and a is any vector, then \[(\mathbf{a}\,.\,\mathbf{b})\,\mathbf{b}+(\mathbf{a}\,.\,\mathbf{c})\,\mathbf{c}+\frac{\mathbf{a}\,.\,(\mathbf{b}\times \mathbf{c})}{|\mathbf{b}\times \mathbf{c}|}\,(\mathbf{b}\times \mathbf{c})=\] [IIT 1996]
  A)
  a done clear
  B)
  b done clear
  C)
  c done clear
  D)
  0 done clear
  View Solution play_arrow

• question_answer30)
  If a, b, c are non-coplanar unit vectors such that \[\mathbf{a}\times (\mathbf{b}\times \mathbf{c})=\frac{\mathbf{b}+\mathbf{c}}{\sqrt{2}}\], then the angle between a and b is [IIT 1995]
  A)
  \[\frac{\pi }{4}\] done clear
  B)
  \[\frac{\pi }{2}\] done clear
  C)
  \[\frac{3\pi }{4}\] done clear
  D)
  \[\pi \] done clear
  View Solution play_arrow

• question_answer31)
  \[[(\mathbf{a}\times \mathbf{b})\times (\mathbf{b}\times \mathbf{c})\,(\mathbf{b}\times \mathbf{c})\times (\mathbf{c}\times \mathbf{a})\,(\mathbf{c}\times \mathbf{a})\times (\mathbf{a}\times \mathbf{b})]=\,\]
  A)
  \[{{[\mathbf{a}\,\,\mathbf{b}\,\,\mathbf{c}]}^{2}}\] done clear
  B)
  \[{{[\mathbf{a}\,\,\mathbf{b}\,\,\mathbf{c}]}^{3}}\] done clear
  C)
  \[{{[\mathbf{a}\,\,\mathbf{b}\,\,\mathbf{c}]}^{4}}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer32)
  Unit vectors a, b and c are coplanar. A unit vector d is perpendicular to them. If \[(\mathbf{a}\times \mathbf{b})\times (\mathbf{c}\times \mathbf{d})=\frac{1}{6}\mathbf{i}-\frac{1}{3}\mathbf{j}+\frac{1}{3}\mathbf{k}\] and the angle between a and b is \[{{30}^{o}}\], then c is    [Roorkee Qualifying 1998]
  A)
  \[\frac{(\mathbf{i}-2\mathbf{j}+2\mathbf{k})}{3}\] done clear
  B)
  \[\frac{(2\mathbf{i}+\mathbf{j}-\mathbf{k})}{3}\] done clear
  C)
  \[\frac{(-\mathbf{i}+2\mathbf{j}-2\mathbf{k})}{3}\] done clear
  D)
  \[\frac{(-\mathbf{i}+2\mathbf{j}+\mathbf{k})}{3}\] done clear
  View Solution play_arrow

• question_answer33)
  The radius of the circular section of the sphere \[|\mathbf{r}|\,=5\]by the plane \[\mathbf{r}\,.\,(\mathbf{i}+\mathbf{j}+\mathbf{k})=3\sqrt{3}\] is [DCE 1999]
  A)
  1 done clear
  B)
  2 done clear
  C)
  3 done clear
  D)
  4 done clear
  View Solution play_arrow

• question_answer34)
  If \[\mathbf{x}\] is parallel to \[\mathbf{y}\] and \[\mathbf{z}\] where \[\mathbf{x}=2\mathbf{i}+\mathbf{j}+\alpha \mathbf{k}\], \[\mathbf{y}=\alpha \mathbf{i}+\mathbf{k}\] and \[\mathbf{z}=5\mathbf{i}-\mathbf{j}\], then \[\alpha \] is equal to [J & K 2005]
  A)
  \[\pm \sqrt{5}\] done clear
  B)
  \[\pm \sqrt{6}\] done clear
  C)
  \[\pm \sqrt{7}\] done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer35)
  The vector c directed along the internal bisector of the angle between the vectors \[\mathbf{a}=7\mathbf{i}-4\mathbf{j}-4\mathbf{k}\] and \[\mathbf{b}=-2\mathbf{i}-\mathbf{j}+2\mathbf{k}\] with \[|\mathbf{c}|\,=5\sqrt{6},\] is   
  A)
  \[\frac{5}{3}\,(\mathbf{i}-7\mathbf{j}+2\mathbf{k})\]                     done clear
  B)
  \[\frac{5}{3}\,(5\mathbf{i}+5\mathbf{j}+2\mathbf{k})\] done clear
  C)
  \[\frac{5}{3}\,(\mathbf{i}+7\mathbf{j}+2\mathbf{k})\]       done clear
  D)
  \[\frac{5}{3}\,(-5\mathbf{i}+5\mathbf{j}+2\mathbf{k})\] done clear
  View Solution play_arrow

• question_answer36)
  The distance of the point \[B\,(\mathbf{i}+2\mathbf{j}+3\mathbf{k})\] from the line which is passing through \[A\,(4\mathbf{i}+2\mathbf{j}+2\mathbf{k})\] and which is parallel to the vector \[\overrightarrow{C}=2\mathbf{i}+3\mathbf{j}+6\mathbf{k}\] is [Roorkee 1993]
  A)
  10 done clear
  B)
  \[\sqrt{10}\] done clear
  C)
  100 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer37)
  Let a, b, c are three non-coplanar vectors such that 

  \[{{\mathbf{r}}_{1}}=\mathbf{a}-\mathbf{b}+\mathbf{c},\,\,{{\mathbf{r}}_{2}}=\mathbf{b}+\mathbf{c}-\mathbf{a},\,\,{{\mathbf{r}}_{3}}=\mathbf{c}+\mathbf{a}+\mathbf{b},\]

  \[\mathbf{r}=2\mathbf{a}-3\mathbf{b}+4\mathbf{c}.\] If \[\mathbf{r}={{\lambda }_{1}}{{\mathbf{r}}_{1}}+{{\lambda }_{2}}{{\mathbf{r}}_{2}}+{{\lambda }_{3}}{{\mathbf{r}}_{3}},\] then

  A)
  \[{{\lambda }_{1}}=7\] done clear
  B)
  \[{{\lambda }_{1}}+{{\lambda }_{3}}=3\] done clear
  C)
  \[{{\lambda }_{1}}+{{\lambda }_{2}}+{{\lambda }_{3}}=4\] done clear
  D)
  \[{{\lambda }_{3}}+{{\lambda }_{2}}=2\] done clear
  View Solution play_arrow

• question_answer38)
  Let \[\mathbf{a}=2\mathbf{i}+\mathbf{j}+\mathbf{k},\,\,\mathbf{b}=\mathbf{i}+2\mathbf{j}-\mathbf{k}\]and a unit vector c be coplanar. If c is perpendicular to a, then c = [IIT 1999; Pb. CET 2003; DCE 2005]
  A)
  \[\frac{1}{\sqrt{2}}(-\mathbf{j}+\mathbf{k})\] done clear
  B)
  \[\frac{1}{\sqrt{3}}(-\mathbf{i}-\mathbf{j}-\mathbf{k})\] done clear
  C)
  \[\frac{1}{\sqrt{5}}\,(\mathbf{i}-2\mathbf{j})\] done clear
  D)
  \[\frac{1}{\sqrt{3}}(\mathbf{i}-\mathbf{j}-\mathbf{k})\] done clear
  View Solution play_arrow

• question_answer39)
  Let p, q, r be three mutually perpendicular vectors of the same magnitude. If a vector x satisfies equation \[\mathbf{p}\times \{(\mathbf{x}-\mathbf{q})\times \mathbf{p}\}+\mathbf{q}\times \{(\mathbf{x}-\mathbf{r})\times \mathbf{q}\}+\mathbf{r}\times \{(\mathbf{x}-\mathbf{p})\times \mathbf{r}\}=0,\] then x is given by [IIT 1997 Cancelled]
  A)
  \[\frac{1}{2}\,(\mathbf{p}+\mathbf{q}-2\mathbf{r})\]          done clear
  B)
  \[\frac{1}{2}(\mathbf{p}+\mathbf{q}+\mathbf{r})\] done clear
  C)
  \[\frac{1}{3}(\mathbf{p}+\mathbf{q}+\mathbf{r})\] done clear
  D)
  \[\frac{1}{3}(2\mathbf{p}+\mathbf{q}-\mathbf{r})\] done clear
  View Solution play_arrow

• question_answer40)
  The point of intersection of \[\mathbf{r}\times \mathbf{a}=\mathbf{b}\times \mathbf{a}\] and \[\mathbf{r}\times \mathbf{b}=\mathbf{a}\times \mathbf{b}\], where \[\mathbf{a}=\mathbf{i}+\mathbf{j}\] and \[\mathbf{b}=2\mathbf{i}-\mathbf{k}\] is [Orissa JEE 2004]
  A)
  \[3\mathbf{i}+\mathbf{j}-\mathbf{k}\] done clear
  B)
  \[3\mathbf{i}-\mathbf{k}\] done clear
  C)
  \[3\mathbf{i}+2\mathbf{j}+\mathbf{k}\] done clear
  D)
  None of these done clear
  View Solution play_arrow