JEE Main & Advanced Mathematics Vector Algebra Question Bank

done Scalar triple product and their applications Total Questions - 63

• question_answer1)
  If a, b, c are three non-coplanar vector, then \[\frac{\mathbf{a}\,.\,\mathbf{b}\times \mathbf{c}}{\mathbf{c}\times \mathbf{a}\,.\,\mathbf{b}}+\frac{\mathbf{b}\,.\,\mathbf{a}\times \mathbf{c}}{\mathbf{c}\,.\,\mathbf{a}\times \mathbf{b}}\]=          [IIT 1985, 86; UPSEAT 2003]
  A)
              0 done clear
  B)
              2 done clear
  C)
              ? 2 done clear
  D)
  None of these done clear
  View Solution play_arrow

• question_answer2)
  If a, b, c be any three non-coplanar vectors, then \[[\mathbf{a}+\mathbf{b}\,\,\,\mathbf{b}+\mathbf{c}\,\,\,\mathbf{c}+\mathbf{a}]=\]     [RPET 1988; MP PET 1990, 02; Kerala (Engg.) 2002]
  A)
              \[|\mathbf{a}\,\mathbf{b}\,\mathbf{c}|\] done clear
  B)
              2\[[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]\] done clear
  C)
              \[{{[\,\mathbf{a}\,\mathbf{b}\,\mathbf{c}\,]}^{2}}\] done clear
  D)
              \[2\,{{[\,\mathbf{a}\,\mathbf{b}\,\mathbf{c}\,]}^{2}}\] done clear
  View Solution play_arrow

• question_answer3)
  If the vectors \[2\mathbf{i}-3\mathbf{j},\,\,\mathbf{i}+\mathbf{j}-\mathbf{k}\] and \[3\mathbf{i}-\mathbf{k}\] form three concurrent edges of a parallelopiped, then the volume of the parallelopiped is        [IIT 1983; RPET 1995; DCE 2001; Kurukshetra CEE 1998; MP PET 2001]
  A)
              8 done clear
  B)
              10 done clear
  C)
              4 done clear
  D)
              14 done clear
  View Solution play_arrow

• question_answer4)
  If a, b, c are any three coplanar unit vectors, then
  A)
              \[\mathbf{a}\,.\,(\mathbf{b}\times \mathbf{c})=1\] done clear
  B)
              \[\mathbf{a}\,.\,(\mathbf{b}\times \mathbf{c})=3\] done clear
  C)
              \[(\mathbf{a}\times \mathbf{b})\,.\,\mathbf{c}=0\] done clear
  D)
              \[(\mathbf{c}\times \mathbf{a})\,.\,\mathbf{b}=1\] done clear
  View Solution play_arrow

• question_answer5)
  If a and b be parallel vectors, then [a c b] =
  A)
              0 done clear
  B)
              1 done clear
  C)
              2 done clear
  D)
              None of these done clear
  View Solution play_arrow

• question_answer6)
  If the vectors \[2\mathbf{i}-\mathbf{j}+\mathbf{k},\,\,\mathbf{i}+2\mathbf{j}-3\mathbf{k}\] and \[3\mathbf{i}+\lambda \mathbf{j}+5\mathbf{k}\] be coplanar, then \[\lambda =\]             [Roorkee 1986; RPET 1999, 02; Kurukshetra CEE 2002]
  A)
              ? 1 done clear
  B)
              ? 2 done clear
  C)
              ? 3 done clear
  D)
              ? 4 done clear
  View Solution play_arrow

• question_answer7)
  If a, b, c are the three non-coplanar vectors and p, q, r are defined by the relations \[\mathbf{p}=\frac{\mathbf{b}\times \mathbf{c}}{[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]},\,\,\mathbf{q}=\frac{\mathbf{c}\times \mathbf{a}}{[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]},\,\,\mathbf{r}=\frac{\mathbf{a}\times \mathbf{b}}{[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]}\] then (a+b) . p +(b+c) . q +(c+a) . r =  [IIT 1988; BIT Mesra 1996; AMU 2002]
  A)
              0 done clear
  B)
              1 done clear
  C)
              2 done clear
  D)
              3 done clear
  View Solution play_arrow

• question_answer8)
  If the points whose position, vectors are \[3\mathbf{i}-2\mathbf{j}-\mathbf{k},\] \[2\mathbf{i}+3\mathbf{j}-4\mathbf{k},\] \[-\mathbf{i}+\mathbf{j}+2\mathbf{k}\]and \[4\mathbf{i}+5\mathbf{j}+\lambda \mathbf{k}\] lie on a plane, then \[\lambda =\]                    [IIT 1986; Pb. CET 2003]
  A)
              \[-\frac{146}{17}\] done clear
  B)
              \[\frac{146}{17}\] done clear
  C)
              \[-\frac{17}{146}\] done clear
  D)
              \[\frac{17}{146}\] done clear
  View Solution play_arrow

• question_answer9)
  If \[\mathbf{p}=\frac{\mathbf{b}\times \mathbf{c}}{[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]},\,\,\mathbf{q}=\frac{\mathbf{c}\times \mathbf{a}}{[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]},\,\,\mathbf{r}=\frac{\mathbf{a}\times \mathbf{b}}{[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]},\,\,\]where a, b, c are three non-coplanar vectors, then the value of \[(\mathbf{a}+\mathbf{b}+\mathbf{c})\,.\,(\mathbf{p}+\mathbf{q}+\mathbf{r})\] is given by   [MNR 1992; UPSEAT 2000]
  A)
              3 done clear
  B)
              2 done clear
  C)
              1 done clear
  D)
              0 done clear
  View Solution play_arrow

• question_answer10)
  The volume of the parallelopiped whose edges are represented by \[-12\mathbf{i}+\alpha \mathbf{k},\,\,3\mathbf{j}-\mathbf{k}\] and \[2\mathbf{i}+\mathbf{j}-15\mathbf{k}\] is 546. Then \[\alpha =\]            [IIT Screening 1989; MNR 1987]
  A)
              3 done clear
  B)
              2 done clear
  C)
              ? 3 done clear
  D)
              ­? 2 done clear
  View Solution play_arrow

• question_answer11)
  Let a, b, c be distinct non-negative numbers. If the vectors \[a\mathbf{i}+a\mathbf{j}+c\mathbf{k},\,\,\mathbf{i}+\mathbf{k}\] and \[c\mathbf{i}+c\mathbf{j}+b\mathbf{k}\] lie in a plane, then c is [IIT 1993; AIEEE 2005]
  A)
              The arithmetic mean of a and b done clear
  B)
            The geometric mean of a and b done clear
  C)
              The harmonic mean of a and b done clear
  D)
              Equal to zero done clear
  View Solution play_arrow

• question_answer12)
  If a, b, c are any three vectors and their inverse are \[{{\mathbf{a}}^{-1}},\,{{\mathbf{b}}^{-1}},\,{{\mathbf{c}}^{-1}}\]and \[[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]\ne 0,\] then \[[{{\mathbf{a}}^{-1}}\,{{\mathbf{b}}^{-1}}\,{{\mathbf{c}}^{-1}}]\] will be [Roorkee 1989]
  A)
              Zero done clear
  B)
              One done clear
  C)
              Non-zero done clear
  D)
              [a b c] done clear
  View Solution play_arrow

• question_answer13)
  If \[\mathbf{a}=\mathbf{i}-\mathbf{j}+\mathbf{k},\,\,\mathbf{b}=\mathbf{i}+2\mathbf{j}-\mathbf{k}\] and \[\mathbf{c}=3\mathbf{i}+p\mathbf{j}+5\mathbf{k}\] are coplanar then the value of p will be     [RPET 1985, 86, 88, 91]
  A)
              ? 6 done clear
  B)
              ? 2 done clear
  C)
              2 done clear
  D)
              6 done clear
  View Solution play_arrow

• question_answer14)
  If \[\mathbf{i},\,\mathbf{j},\,\mathbf{k}\] are the unit vectors and mutually perpendicular, then \[[\mathbf{i}\,\mathbf{k}\,\mathbf{j}]\] is equal to                            [RPET 1986]
  A)
              0 done clear
  B)
              ? 1 done clear
  C)
              1 done clear
  D)
              None of these done clear
  View Solution play_arrow

• question_answer15)
  If three vectors \[\mathbf{a}=12\mathbf{i}+4\mathbf{j}+3\mathbf{k},\,\,\]\[\mathbf{b}=8\mathbf{i}-12\mathbf{j}-9\mathbf{k}\] and \[\mathbf{c}=33\mathbf{i}-4\mathbf{j}-24\mathbf{k}\] represents a cube, then its volume will be [Roorkee 1988]
  A)
              616 done clear
  B)
              308 done clear
  C)
              154 done clear
  D)
              None of these done clear
  View Solution play_arrow

• question_answer16)
  If \[\mathbf{a}=2\mathbf{i}+\mathbf{j}-\mathbf{k},\,\,\mathbf{b}=\mathbf{i}+2\mathbf{j}+\mathbf{k}\] and \[\mathbf{c}=\mathbf{i}-\mathbf{j}+2\mathbf{k},\] then \[\mathbf{a}\,.\,(\mathbf{b}\times \mathbf{c})=\]                                    [RPET 1989, 2001]
  A)
              6 done clear
  B)
              10 done clear
  C)
              12 done clear
  D)
              24 done clear
  View Solution play_arrow

• question_answer17)
  Three concurrent edges OA, OB, OC of a parallelopiped are represented by three vectors \[2\mathbf{i}+\mathbf{j}-\mathbf{k},\,\,\mathbf{i}+2\mathbf{j}+3\mathbf{k}\] and \[-3\mathbf{i}-\mathbf{j}+\mathbf{k},\] the volume of the solid so formed in cubic unit is [Kurukshetra CEE 1998]
  A)
              5 done clear
  B)
              6 done clear
  C)
              7 done clear
  D)
              8 done clear
  View Solution play_arrow

• question_answer18)
  If \[\mathbf{x}\,.\,\mathbf{a}=0,\,\,\mathbf{x}\,.\,\mathbf{b}=0\] and \[\mathbf{x}\,.\,\mathbf{c}=0\] for some non-zero vector x, then the ture statement is [IIT 1983; Karnataka CET 2002]
  A)
              \[[\mathbf{a}\,\,\mathbf{b}\,\,\mathbf{c}]=0\] done clear
  B)
              \[[\mathbf{a}\,\,\mathbf{b}\,\,\mathbf{c}]\ne 0\] done clear
  C)
              \[[\mathbf{a}\,\,\mathbf{b}\,\,\mathbf{c}]=1\] done clear
  D)
              None of these done clear
  View Solution play_arrow

• question_answer19)
  If the given vectors \[(-bc,\,{{b}^{2}}+bc,\,{{c}^{2}}+bc),\] \[({{a}^{2}}+ac,\,-ac,\,{{c}^{2}}+ac)\] and \[({{a}^{2}}+ab,\,{{b}^{2}}+ab,\,-ab)\] are coplanar, where none of a, b and c is zero, then
  A)
              \[{{a}^{2}}+{{b}^{2}}+{{c}^{2}}=1\] done clear
  B)
              \[bc+ca+ab=0\] done clear
  C)
              \[a+b+c=0\] done clear
  D)
              \[{{a}^{2}}+{{b}^{2}}+{{c}^{2}}=bc+ca+ab\] done clear
  View Solution play_arrow

• question_answer20)
  If a,b,c are three coplanar vectors, then \[[\mathbf{a}+\mathbf{b}\,\,\mathbf{b}+\mathbf{c}\,\,\mathbf{c}+\mathbf{a}]=\] [MP PET 1995]
  A)
              [a b c] done clear
  B)
              2 [a b c] done clear
  C)
              3 [a b c] done clear
  D)
              0 done clear
  View Solution play_arrow

• question_answer21)
  \[[\mathbf{a}\,\,\mathbf{b}\,\,\mathbf{a}\times \mathbf{b}]\] is equal to
  A)
              \[|\,\,\mathbf{a}\times \mathbf{b}|\] done clear
  B)
              \[|\,\,\mathbf{a}\times \mathbf{b}{{|}^{2}}\] done clear
  C)
              0 done clear
  D)
              None of these done clear
  View Solution play_arrow

• question_answer22)
  If \[\mathbf{a}\,.\,\mathbf{i}=4,\] then \[(\mathbf{a}\times \mathbf{j})\,.\,(2\mathbf{j}-3\mathbf{k})=\] [EAMCET 1994]
  A)
              12 done clear
  B)
              2 done clear
  C)
              0 done clear
  D)
              ? 12 done clear
  View Solution play_arrow

• question_answer23)
  If the vectors \[2\mathbf{i}-3\mathbf{j}+4\mathbf{k},\,\,\mathbf{i}+2\mathbf{j}-\mathbf{k}\] and \[x\mathbf{i}-\mathbf{j}+2\mathbf{k}\] are coplanar, then \[x=\] [EAMCET 1994]
  A)
              \[\frac{8}{5}\] done clear
  B)
              \[\frac{5}{8}\] done clear
  C)
              0 done clear
  D)
              1 done clear
  View Solution play_arrow

• question_answer24)
  Volume of the parallelopiped whose coterminous edges are \[2\mathbf{i}-3\mathbf{j}+4\mathbf{k},\,\,\mathbf{i}+2\mathbf{j}-2\mathbf{k},\,\,3\mathbf{i}-\mathbf{j}+\mathbf{k},\] is [EAMCET 1993]
  A)
              5 cubic unit done clear
  B)
              6 cubic unit done clear
  C)
              7 cubic unit done clear
  D)
              8 cubic unit done clear
  View Solution play_arrow

• question_answer25)
  If \[\mathbf{a}=3\mathbf{i}-\mathbf{j}+2\mathbf{k},\] \[\mathbf{b}=2\mathbf{i}+\mathbf{j}-\mathbf{k},\] then \[\mathbf{a}\times (\mathbf{a}\,.\,\mathbf{b})=\] [Karnataka CET 1994]
  A)
              3a done clear
  B)
              \[3\sqrt{14}\] done clear
  C)
              0 done clear
  D)
              None of these done clear
  View Solution play_arrow

• question_answer26)
  \[\mathbf{i}\,.\,(\mathbf{j}\times \mathbf{k})+\mathbf{j}\,.\,(\mathbf{k}\times \mathbf{i})+\mathbf{k}\,.\,(\mathbf{i}\times \mathbf{j})=\]            [Karnataka CET 1994]
  A)
              1 done clear
  B)
              3 done clear
  C)
              ? 3 done clear
  D)
              0 done clear
  View Solution play_arrow

• question_answer27)
  If \[\mathbf{a}=\mathbf{i}-2\mathbf{j}+3\mathbf{k}\] and \[\mathbf{b}=3\mathbf{i}+\mathbf{j}+2\mathbf{k},\] then the unit vector perpendicular to a and b is [MP PET 1996]
  A)
              \[\frac{\mathbf{i}+\mathbf{j}+\mathbf{k}}{\sqrt{3}}\] done clear
  B)
              \[\frac{\mathbf{i}-\mathbf{j}+\mathbf{k}}{\sqrt{3}}\] done clear
  C)
              \[\frac{-\mathbf{i}+\mathbf{j}+\mathbf{k}}{\sqrt{3}}\] done clear
  D)
              \[\frac{\mathbf{i}-\mathbf{j}-\mathbf{k}}{\sqrt{3}}\] done clear
  View Solution play_arrow

• question_answer28)
  If \[\mathbf{a}=-3\mathbf{i}+7\mathbf{j}+5\mathbf{k},\] \[\mathbf{b}=-3\mathbf{i}+7\mathbf{j}-3\mathbf{k}\], \[\mathbf{c}=7\mathbf{i}-5\mathbf{j}-3\mathbf{k}\] are the three coterminous edges of a parallelopiped, then its volume is [MP PET 1996]
  A)
              108 done clear
  B)
              210 done clear
  C)
              272 done clear
  D)
              308 done clear
  View Solution play_arrow

• question_answer29)
  \[\mathbf{a}\,.\,(\mathbf{a}\times \mathbf{b})=\] [MP PET 1996]
  A)
              b . b done clear
  B)
              \[{{a}^{2}}b\] done clear
  C)
              0 done clear
  D)
              \[{{a}^{2}}+ab\] done clear
  View Solution play_arrow

• question_answer30)
  If three conterminous edges of a parallelopiped are represented by \[\mathbf{a}-\mathbf{b},\,\,\mathbf{b}-\mathbf{c}\] and \[\mathbf{c}-\mathbf{a}\], then its volume is                                       [MP PET 1999; Pb. CET 2003]
  A)
              [a b c] done clear
  B)
              2 [a b c] done clear
  C)
              \[\,{{[\mathbf{a}\,\,\mathbf{b}\,\,\mathbf{c}]}^{2}}\] done clear
  D)
              0 done clear
  View Solution play_arrow

• question_answer31)
  For three vectors u, v, w which of the following expressions is not equal to any of the remaining three         [IIT 1998]
  A)
              \[\mathbf{u}\,.\,(\mathbf{v}\times \mathbf{w})\] done clear
  B)
              \[\,(\mathbf{v}\times \mathbf{w})\,.\,\mathbf{u}\,\] done clear
  C)
              \[\mathbf{v}\,.\,(\mathbf{u}\times \mathbf{w})\] done clear
  D)
              \[(\mathbf{u}\times \mathbf{v})\,.\,\mathbf{w}\] done clear
  View Solution play_arrow

• question_answer32)
  Which of the following expressions are meaningful [IIT 1998; RPET 2001]
  A)
              \[\mathbf{u}\,.\,(\mathbf{v}\times \mathbf{w})\] done clear
  B)
              \[(\mathbf{u}\,.\,\mathbf{v})\,.\,\mathbf{w}\] done clear
  C)
              \[(\mathbf{u}\,.\,\mathbf{v})\,\mathbf{w}\] done clear
  D)
              \[\mathbf{u}\times (\mathbf{v}\,.\,\mathbf{w})\] done clear
  View Solution play_arrow

• question_answer33)
  If \[\mathbf{a,}\,\mathbf{b,}\,\mathbf{c}\] are non-coplanar vectors and \[\mathbf{d}=\lambda \mathbf{a}+\mu \,\mathbf{b}+\nu \mathbf{c},\] then \[\lambda \] is equal to [Roorkee 1999]
  A)
              \[\frac{[\mathbf{d}\,\mathbf{b}\,\mathbf{c}]}{[\mathbf{b}\,\mathbf{a}\,\mathbf{c}]}\] done clear
  B)
              \[\frac{[\mathbf{b}\,\mathbf{c}\,\mathbf{d}]}{[\mathbf{b}\,\mathbf{c}\,\mathbf{a}]}\] done clear
  C)
              \[\frac{[\mathbf{b}\,\mathbf{d}\,\mathbf{c}]}{[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]}\] done clear
  D)
              \[\frac{[\mathbf{c}\,\mathbf{b}\,\mathbf{d}\,]}{[\mathbf{a}\,\mathbf{b}\,\mathbf{c}]}\] done clear
  View Solution play_arrow

• question_answer34)
  If vectors \[\vec{A}=2\mathbf{i}+3\mathbf{j}+4\mathbf{k}\], \[\vec{B}=\mathbf{i}+\mathbf{j}+5\mathbf{k}\], and \[\vec{C}\] form a left handed system, then \[\vec{C}\] is                      [Roorkee 1999]
  A)
              11i ? 6j ? k done clear
  B)
              ? 11i + 6j + k done clear
  C)
              11i ? 6j + k done clear
  D)
              ? 11i + 6j ? k done clear
  View Solution play_arrow

• question_answer35)
  What will be the volume of that parallelopiped whose sides are a = i ? j + k, b = i ? 3j + 4k and c = 2i ? 5j +  3k [UPSEAT 1999]
  A)
              5 unit done clear
  B)
              6 unit done clear
  C)
              7 unit done clear
  D)
              8 unit done clear
  View Solution play_arrow

• question_answer36)
  Given vectors a, b, c such that \[\mathbf{a}\,.(\mathbf{b}\times \mathbf{c})\]\[=\lambda \ne 0,\,\] the value of \[(\mathbf{b}\times \mathbf{c})\,.\,(\mathbf{a}+\mathbf{b}+\mathbf{c})/\lambda \]  is                            [AMU 1999]
  A)
              3          done clear
  B)
              1 done clear
  C)
              \[-3\lambda \] done clear
  D)
              \[3/\lambda \] done clear
  View Solution play_arrow

• question_answer37)
  If \[a,\,b\] and c are unit coplanar vectors then the scalar triple product \[[2a-b\,\,2b-c\,\,2c-a]\] is equal to    [IIT Screening 2000; Kerala (Engg.) 2005]
  A)
              0 done clear
  B)
              1 done clear
  C)
              \[-\sqrt{3}\] done clear
  D)
              \[\sqrt{3}\] done clear
  View Solution play_arrow

• question_answer38)
  If the vectors \[i+3j-2k\], \[2i-j+4k\] and \[3i+2j+xk\] are coplanar, then the value of x is                 [Karnataka CET 2000]
  A)
              ? 2 done clear
  B)
              2 done clear
  C)
              1 done clear
  D)
              3 done clear
  View Solution play_arrow

• question_answer39)
  The value of [a ? b b ? c c ? a], where \[|a|\,=1\], \[|b|\,=5\] and \[|c|\,=3\] is                           [RPET 2000]
  A)
              0 done clear
  B)
              1 done clear
  C)
              2 done clear
  D)
              4 done clear
  View Solution play_arrow

• question_answer40)
  \[a=i+j+k,\,\mathbf{b}=2i-4k,\,c=i+\lambda \,j+3k\] are coplanar, then the value of \[\lambda \] is [MP PET 2000]
  A)
              5/2 done clear
  B)
              3/5 done clear
  C)
              7/3 done clear
  D)
              None of these done clear
  View Solution play_arrow

• question_answer41)
  Let  \[\overrightarrow{A}=i+j+k\], \[\overrightarrow{B}=i,\,\overrightarrow{C}={{C}_{1}}i+{{C}_{2}}j+{{C}_{3}}k\]. If \[{{C}_{2}}=-1\], and \[{{C}_{3}}=1\], then to make three vectors coplanar         [AMU 2000]
  A)
              \[{{C}_{1}}=0\] done clear
  B)
              \[{{C}_{1}}=1\] done clear
  C)
              \[{{C}_{1}}=2\] done clear
  D)
              No value of \[{{C}_{1}}\] can be found done clear
  View Solution play_arrow

• question_answer42)
  Let \[a=i-k,\,\,\,b=xi+j+(1-x)\,k\],\[c=yi+xj+(1+x-y)k\]. Then \[[a\,\,b\,\,c]\] depends on [IIT Screening 2001; AIEEE 2005]
  A)
              Only x done clear
  B)
              Only y done clear
  C)
              Neither x nor y done clear
  D)
              Both x and y done clear
  View Solution play_arrow

• question_answer43)
  If \[\mathbf{a}=3\mathbf{i}-2\mathbf{j}+2\mathbf{k},\,\,\,\mathbf{b}=6\mathbf{i}+4\mathbf{j}-2\mathbf{k}\] and \[\mathbf{c}=3\mathbf{i}-2\mathbf{j}-4\mathbf{k}\], then \[\mathbf{a}\,.\,\,(\mathbf{b}\times \mathbf{c})\] is [Karnataka CET 2001]
  A)
              122 done clear
  B)
              ? 144 done clear
  C)
              120 done clear
  D)
              ? 120 done clear
  View Solution play_arrow

• question_answer44)
  \[(\mathbf{a}+\mathbf{b})\,.\,(\mathbf{b}+\mathbf{c})\times (\mathbf{a}+\mathbf{b}+\mathbf{c})=\]                                    [EAMCET 2002]
  A)
              ? [a b c] done clear
  B)
              [a b c] done clear
  C)
              0 done clear
  D)
              2[a b c] done clear
  View Solution play_arrow

• question_answer45)
  a.(b × c) is equal to [RPET 2001]
  A)
              b.(a × c) done clear
  B)
              c.(b × a) done clear
  C)
                  It is obvious. done clear
  D)
              None of these done clear
  View Solution play_arrow

• question_answer46)
  If \[a,\,b,\,c\] are vectors such that \[[a\,b\,c\,]=4\], then \[[a\times b\,\,b\times c\,\,c\times a]\] = [AIEEE 2002]
  A)
              16 done clear
  B)
              64 done clear
  C)
              4 done clear
  D)
              8 done clear
  View Solution play_arrow

• question_answer47)
  The volume of the parallelopiped whose conterminous edges are \[i-j+k,\,\,2i-4j+5k\]and \[3i-5j+2k\] is [Kerala (Engg.) 2002]
  A)
              4 done clear
  B)
              3 done clear
  C)
              2 done clear
  D)
              8          done clear
  View Solution play_arrow

• question_answer48)
  \[[i\,\,k\,\,j]+[k\,\,j\,\,i]+[j\,\,k\,\,i]\] [UPSEAT 2002]
  A)
              1 done clear
  B)
              3 done clear
  C)
              ? 3 done clear
  D)
              ? 1       done clear
  View Solution play_arrow

• question_answer49)
  If u, v and w are three non-coplanar vectors, then \[(u+v-w)\,.\,[(u-v)\times (v-w)]\] equals [AIEEE 2003; DCE 2005]
  A)
              0 done clear
  B)
              \[u\,.\,(v\times w)\] done clear
  C)
              \[u\,.\,(w\times v)\] done clear
  D)
              \[3u\,.\,(v\times w)\] done clear
  View Solution play_arrow

• question_answer50)
  \[a\,.\,[(b+c)\times (a+b+c)]\] is equal to [IIT 1981; UPSEAT 2003; RPET 1988, 2002; MP PET 2004]
  A)
              [a b c] done clear
  B)
              2[a b c] done clear
  C)
              3[a b c] done clear
  D)
              0 done clear
  View Solution play_arrow

• question_answer51)
  If the vectors \[4i+11j+mk,\,7i+2j+6k\] and \[i+5j+4k\] are coplanar, then m is                            [Karnataka CET 2003]
  A)
              38 done clear
  B)
              0 done clear
  C)
              10 done clear
  D)
              ? 10 done clear
  View Solution play_arrow

• question_answer52)
  Vector coplanar with vectors i + j and j + k and parallel to the vector 2i ? 2j ? 4k, is                      [Roorkee 2000]
  A)
              i ? k done clear
  B)
              i ? j ? 2k         done clear
  C)
              i + j ? k done clear
  D)
              3i + 3j ? 6k done clear
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• question_answer53)
  The value of l for which the four points \[2\mathbf{i}+3\mathbf{j}-\mathbf{k},\] \[\mathbf{i}+2\mathbf{j}+3\mathbf{k}\], \[3\mathbf{i}+4\mathbf{j}-2\mathbf{k},\,\,\mathbf{i}-\lambda \mathbf{j}+6\mathbf{k}\] are coplanar                                     [MP PET 2004]
  A)
              8 done clear
  B)
              0 done clear
  C)
              ? 2 done clear
  D)
              6 done clear
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• question_answer54)
  If \[\mathbf{a},\,\,\mathbf{b},\,\,\mathbf{c}\] are non-coplanar vectors and l is a real number, then the vectors \[\mathbf{a}+2\mathbf{b}+3\mathbf{c},\,\lambda \,\mathbf{b}+4\mathbf{c}\] and \[(2\lambda -1)\mathbf{c}\] are non-coplanar for [AIEEE 2004]
  A)
              No value of l done clear
  B)
              All except one value of l done clear
  C)
              All except two values of l done clear
  D)
              All values of l done clear
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• question_answer55)
  Let \[\mathbf{a},\,\mathbf{b}\] and \[\mathbf{c}\] be three vectors. Then scalar triple product x is equal to                                     [UPSEAT 2004]
  A)
              done clear
  B)
              done clear
  C)
              done clear
  D)
              done clear
  View Solution play_arrow

• question_answer56)
  . If \[\mathbf{a}\,.\,\mathbf{b}=\mathbf{b}\,.\,\mathbf{c}=\mathbf{c}\,.\,\mathbf{a}=0\] then the value of  [a b c] is equal to                                        [Pb. CET 2000]
  A)
              1 done clear
  B)
              ? 1 done clear
  C)
              \[|\mathbf{a}||\mathbf{b}||\mathbf{c}|\] done clear
  D)
              0 done clear
  View Solution play_arrow

• question_answer57)
  If \[\mathbf{a}=\mathbf{i}+\mathbf{j}-\mathbf{k},\,\,\mathbf{b}=2\mathbf{i}+3\mathbf{j}+\mathbf{k}\] and \[\mathbf{c}=\mathbf{i}+\alpha \mathbf{j}\] are coplanar vectors, the value of \[\alpha \] is   [UPSEAT 2004]
  A)
              \[-\frac{4}{3}\] done clear
  B)
              \[\frac{3}{4}\] done clear
  C)
              \[\frac{4}{3}\] done clear
  D)
              2 done clear
  View Solution play_arrow

• question_answer58)
  Out of the following which one is not true        [Orissa JEE 2004]
  A)
              \[\mathbf{a}\,.\,(\mathbf{b}\times \mathbf{c})\] done clear
  B)
              \[(\mathbf{b}\times \mathbf{c})\,.\,\mathbf{a}\] done clear
  C)
              \[(\mathbf{a}\times \mathbf{b})\,.\,\mathbf{c}\] done clear
  D)
              \[(\mathbf{a}.\mathbf{c})\,\times \,\mathbf{b}\] done clear
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• question_answer59)
  If \[\mathbf{a}\] is perpendicular to \[\mathbf{b}\]and \[\mathbf{c},|\mathbf{a}|=2,|\mathbf{b}|=3\], \[|\mathbf{c}|=4\] and the angle between \[\mathbf{b}\] and \[\mathbf{c}\]is \[\frac{2\pi }{3}\], then \[[\mathbf{a}\ \mathbf{b}\ \mathbf{c}]\] is equal to                          [Kerala (Engg.) 2005]
  A)
              \[4\sqrt{3}\] done clear
  B)
              \[6\sqrt{3}\] done clear
  C)
              \[12\sqrt{3}\] done clear
  D)
              \[18\sqrt{3}\] done clear
  E)
              \[8\sqrt{3}\] done clear
  View Solution play_arrow

• question_answer60)
  If \[\mathbf{a},\,\mathbf{b},\,\mathbf{c}\] are non-coplanar vectors and \[\lambda \] is a real number then \[[\lambda (\mathbf{a}+\mathbf{b})\,\,\,\,{{\lambda }^{2}}\mathbf{b}\,\,\,\,\,\lambda \mathbf{c}]=\left[ \mathbf{a}\,\,\mathbf{b}+\mathbf{c}\,\,\mathbf{b} \right]\] for       [AIEEE 2005]
  A)
              Exactly three values of \[\lambda \]      done clear
  B)
              Exactly two values of \[\lambda \] done clear
  C)
              Exactly one value of \[\lambda \] done clear
  D)
              No value of \[\lambda \] done clear
  View Solution play_arrow

• question_answer61)
  If the vectors \[2\mathbf{i}+\mathbf{j}-\mathbf{k},\,-\mathbf{i}+2\mathbf{j}+\lambda \mathbf{k}\] and  \[-5\mathbf{i}+2\mathbf{j}-\mathbf{k}\] are coplanar, then the value of \[\lambda \] is equal             [J & K 2005]
  A)
              ? 13 done clear
  B)
              13/9 done clear
  C)
              ? 13/9 done clear
  D)
              ? 9/13 done clear
  View Solution play_arrow

• question_answer62)
  If a,b,c are three non-zero, non-coplanar vectors and \[{{b}_{1}}=b-\frac{b.a}{|a{{|}^{2}}}a,\,{{b}_{2}}=b+\frac{b.a}{|a{{|}^{2}}}a\],\[{{c}_{1}}=c-\frac{c.a}{|a{{|}^{2}}}a-\frac{c.b}{|b{{|}^{2}}}b\], \[{{c}_{2}}=c-\frac{c.a}{|a{{|}^{2}}}a\frac{c.{{b}_{1}}}{|{{b}_{1}}{{|}^{2}}}{{b}_{1}}\], \[{{c}_{3}}=c-\frac{c.a}{|a{{|}^{2}}}a\frac{c.{{b}_{2}}}{|{{b}_{2}}{{|}^{2}}}{{b}_{2}}\], \[{{c}_{4}}=a-\frac{c.a}{|a{{|}^{2}}}a\]. Then which of the following is a set of mutually orthogonal vectors is                        [IIT Screening 2005]
  A)
              \[\{\mathbf{a},\,{{\mathbf{b}}_{1}},\,{{c}_{1}}\}\] done clear
  B)
              \[\{\mathbf{a},\,{{\mathbf{b}}_{1}},\,{{c}_{2}}\}\] done clear
  C)
              \[\{\mathbf{a},\,{{\mathbf{b}}_{2}},\,{{c}_{3}}\}\] done clear
  D)
              \[\{\mathbf{a},\,{{\mathbf{b}}_{2}},\,{{c}_{4}}\}\] done clear
  View Solution play_arrow

• question_answer63)
  If a vector a lie in the plane b and g then which is correct                                     [Orissa JEE 2005]
  A)
              \[[\alpha \,\,\beta \,\,\gamma ]=0\] done clear
  B)
              \[[\alpha \,\,\beta \,\,\gamma ]=1\] done clear
  C)
              \[[\alpha \,\,\beta \,\,\gamma ]=3\] done clear
  D)
              \[[\beta \,\,\gamma \,\,\alpha ]=1\] done clear
  View Solution play_arrow