Mathematics - Vector Algebra
Exam Duration: 45 Mins Total Questions : 30
If, \(\vec { a } ,\vec { b } ,\vec { c } \) are collinear, then value of \(\vec { a } \times \vec { b } +\vec { b } \times \vec { c } +\vec { c } \times \vec { a } \) is
- (a)
\(\vec { a } +\vec { b } \)
- (b)
\(\vec { b } +\vec { c } \)
- (c)
\(\vec { c } +\vec { a } \)
- (d)
0
Three concurrent edges OA, OB, OC of a parallelopiped are represented respectively by the three vectors
2i+j-k, i+2j+3k, -3-j+k
The area of the face made by OA and OB is
- (a)
\(\sqrt { 74 } \)
- (b)
\(\sqrt { 77 } \)
- (c)
\(\sqrt {80 } \)
- (d)
\(\sqrt { 83 } \)
Let \(\vec { a } =2\hat { i } +\hat { j } -2\hat { k } \quad and\quad \hat { b } =\hat { i } +\hat { j } .\) if \(\vec { c } \) is a vector such that \(\vec { a } .\vec { c } =\left| \vec { c } \right| ,\quad \left| \vec { c } -\vec { a } \right| =2\sqrt { 2 } \) and the angle between \((\vec { a } \times \vec { b } )\quad and\quad \vec { c } \) is \({ 30 }^{ \circ }\), then value of \(\left| (\vec { a } \times \vec { b } )\times \vec { c } \right| ,\) is
- (a)
\(2\over 3\)
- (b)
\(3\over 2\)
- (c)
2
- (d)
3
If \(a=\hat { i } +2\hat { j } +3\hat { k } ,b=-\hat { i } +2\hat { j } +\hat { k } and\quad c=3\hat { i } +\hat { j } ,\) then p such that a+pb is at right angle to c will be
- (a)
7
- (b)
9
- (c)
3
- (d)
5
Observe the following columns
Column I | Column II | ||
A | If |a| = |b| = |c|, angle between each pair of vectors is \(\frac { \pi }{ 3 } \) and \(|a+b+c|=\sqrt { 6 } \) , then 2|a| is equal to | p. | 3 |
B | If a is perpendicular to b+c, b is perpendicular to c+a, c is perpendicular to a+b,|a|=2, |b|=3 and |c|=6, then |a+b+c|-2 is equal to | q. | 2 |
C | \(a=2\hat { i } +3\hat { j } -\hat { k } ,b=\hat { i } +2\hat { j } -4\hat { k } \) and \(d=3\hat { i } +2\hat { j } +\hat { k } \) , then \(\frac { 1 }{ 7 } (a\times b).(c\times d)\) is equal to | r. | 4 |
D | If |a| = |b| =|c| =2 and a.b=b.c=c.a=2, then [a b c]cos 45o is equal to | s. | 5 |
- (a)
A B C D r s p q - (b)
A B C D q p r s - (c)
A B C D q s p r - (d)
None of the above
A unit vector perpendicular to the plane defined by the vectors defined by the vectors \(a=2\hat { i } -6\hat { j } -3\hat { k } \quad and\quad b=4\hat { i } +3\hat { j } -\hat { k } \) is
- (a)
\(\frac { 1 }{ 7 } \left( 3\hat { i } -2\hat { j } -6\hat { k } \right) \)
- (b)
\(\frac { 1 }{ 7 } \left( 3\hat { i } +2\hat { j } -6\hat { k } \right) \)
- (c)
\(\frac { 1 }{ 7 } \left( -3\hat { i } +2\hat { j } -6\hat { k } \right) \)
- (d)
\(None\quad of\quad the\quad above\)
The vectors a and b are not perpendicular and c and d are two vectors satisfying bXc = bXd and a.d=0. Then, the vector d is equal to
- (a)
\(c+\left( \frac { a.c }{ a.b } \right) b\)
- (b)
\(b+\left( \frac { b.c }{ a.b } \right) c\)
- (c)
\(c+\left( \frac { a.c }{ a.b } \right) b\)
- (d)
\(b-\left( \frac { b.c }{ a.b } \right) c\)
Let a,b and c be distinct non - negative numbers. If the vectors \(a\hat { i } +a\hat { j } +c\hat { k } ,\hat { i } +\hat { k } \quad and\quad c\hat { i } +c\hat { j } +b\hat { k } \) lie in a plane, then c is
- (a)
the harmonic mean of a and b
- (b)
equal to zero
- (c)
the arithmetic mean of a and b
- (d)
the geometric mean of a and b
If \(\vec { a } \times \vec { b } =\vec { c } \) and \(\vec { b } \times \vec { c } =\vec { a } \), then
- (a)
\(\left| \vec { a } \right| =1,\left| \vec { b } \right| =\left| \vec { c } \right| \)
- (b)
\(\left| \vec { c } \right| =1,\left| \vec { a } \right| =1\)
- (c)
\(\left| \vec { b } \right| =2,\left| \vec { b } \right| =2\left| \vec { a } \right| \)
- (d)
\(\left| \vec { b } \right| =1,\left| \vec { c } \right| =\left| \vec { a } \right| \)
Let \(\vec {a},\vec {b},\vec {c}\)be three vectors such |\(\vec {a}\)|+|\(\vec {b}\)|+|\(\vec {c}\)|=4 and angle between \(\vec {a}\) and \(\vec {b}\) is \(\pi\)/3, angle between \(\vec {b}\) and \(\vec {c}\) is \(\pi\)/3 and angle between \(\vec {c}\) and is \(\pi\)/3. The height of the parallelopiped whose adjacent edges are represented by the vectors \(\vec {a}\), \(\vec {b}\) and \(\vec {c}\) is
- (a)
\(4\sqrt { \frac { 2 }{ 3 } } \)
- (b)
\(3\sqrt { \frac { 2 }{ 3 } } \)
- (c)
\(4\sqrt { \frac { 3 }{ 2 } } \)
- (d)
\(4\sqrt { \frac { 3 }{ 2 } } \)
If \(\vec { a } ,\quad \vec { b } \quad and\quad \vec { c } \) be any three non-coplanar vectors. Then system \(\vec { a' } ,\) \(\vec { b' } \) and \(\vec { c' } \) which satisfies \(\vec { a } .\vec { a' } =\vec { b } .\vec { b' } =\vec { c } .\vec { c' } =1\) and \(\vec { a } .\vec { b' } =\vec { a } .\vec { c' } =\vec { b } .\vec { a' } =\vec { b } .\vec { c' } =\vec { c } .\vec { a' } =\vec { c } .\vec { b' } =0\) is called the reciprocal system to the vectors \(\vec { a } ,\quad \vec { b } \quad and\quad \vec { c } \) . The value of \(\left[ \vec { a } \vec { b } \vec { c } \right] \left\{ (\vec { a' } .\vec { a' } )\vec { a } +(\vec { a' } .\vec { b' } )\vec { b } +(\vec { a' } .\vec { c' } )\vec { c } \right\} \)is
- (a)
0
- (b)
\(\vec { a } \times \vec { b } \)
- (c)
\(\vec { b } \times \vec { c } \)
- (d)
\({ \left[ \vec { a } \vec { b } \vec { c } \right] }^{ -2 }\)
If are threee unit vectors, such that is also a unit vectors are angles between the vectors and respectively then among and
- (a)
all are acute angles
- (b)
all are right angles
- (c)
at least one is obtuse angle
- (d)
none of these
If \(\left| \vec { a } \right| =3\left| \vec { b } \right| =4\) and \(\left| \vec { a } +\vec { b } \right| =5\), then \(\left| \vec { a } -\vec { b } \right| \) is Equal to
- (a)
3
- (b)
4
- (c)
5
- (d)
6
If \((\vec { a } \times \vec { b } )^{ 2 }+(\vec { a } .\vec { b } )^{ 2 }=144\left| \vec { a } \right| =4\quad \left| \vec { b } \right| \)is Equal to
- (a)
16
- (b)
8
- (c)
3
- (d)
12
If \(\vec { a } \) is a unit vector such that \(\vec { a } \times (\hat { i } +\hat { j } +\hat { k } )=\hat { i } -\hat { k } \) then \(\vec { a } \) is equal to
- (a)
\(-\frac { 1 }{ 3 } (2\hat { i } +\hat { j } +\hat { 2k } )\quad \)
- (b)
\(\hat { j } \)
- (c)
\(\frac { 1 }{ 3 } (2\hat { i } +\hat { j } +\hat { 2k } )\)
- (d)
\(i\)
The position vectors of the points A, B and C are \(\hat { i } +\hat { j } +\hat { k } +,\hat { i } +5\hat { j } -\hat { k } \) and \(2\hat { i } +3\hat { j } +5\hat { k } \) respectively. The greatest angle of the triangle ABC is
- (a)
900
- (b)
1350
- (c)
\(\cos ^{ -1 }{ \left( \frac { 2 }{ 3 } \right) } \)
- (d)
\(\cos ^{ -1 }{ \left( \frac { 5 }{ 7 } \right) } \)
Let \(\overrightarrow { u } ,\overrightarrow { v } ,\overrightarrow { w } \) be three unit vectors such that \(\overrightarrow { u } +\overrightarrow { v } +\overrightarrow { w } =\overrightarrow { a } ,\overrightarrow { a } .\overrightarrow { u } =\frac { 3 }{ 2 } ,\overrightarrow { a } .\overrightarrow { v } =\frac { 7 }{ 4 } \)and \(|\overrightarrow { a } |=2,\)then
- (a)
\(\overrightarrow { u } .\overrightarrow { v } =\frac { 3 }{ 4 } \)
- (b)
\(\overrightarrow { v } .\overrightarrow { w } =0\)
- (c)
\(\overrightarrow { u } .\overrightarrow { w } =-\frac { 1 }{ 4 } \)
- (d)
none of these
If \(\left( \cfrac { 1 }{ 2 } ,\cfrac { 1 }{ 3 } ,n \right) \) are the direction consines of a line, then the value of n is
- (a)
\(\cfrac { \sqrt { 23 } }{ 6 } \)
- (b)
\(\cfrac { 23 }{ 6 } \)
- (c)
\(\cfrac { 2 }{ 3 } \)
- (d)
\(\cfrac { 3 }{ 2 } \)
The vectors \(\vec { a } =x\hat { i } -2\hat { j } +5\hat { k } \) and \(\vec { b } =\hat { i } -y\hat { j } +z\hat { k } \) are collinear, if
- (a)
x = 1, y = -2, z = -5
- (b)
x = 1/2, y = -4, z = -10
- (c)
x = -1/2, y=4, z = 10
- (d)
All of these
Let \(\vec { a } ,\vec { b } and\quad \vec { c } \) be three non-zero vectors such that no two of these are collinear. If the vectors \(\vec { a } ,2\vec { b } \) is collinear with \(\vec { c } \) and \(\vec { b } +3\vec { c } \) ois collinear with \(\vec { a } \) then \(\vec { a } +2\vec { b } +6\vec { c } \) equal to
- (a)
\(\vec { a } \)
- (b)
\(\vec { b } \)
- (c)
\(\vec { c } \)
- (d)
\(\vec { 0 } \)
The angle between two vectors \(\vec { a } \) and \(\vec { b } \) with magnitudes \(\sqrt { 3 } \) and ,respectively and \(\vec { a. } \vec { b } =2\sqrt { 3 } \) is
- (a)
\(\cfrac { \pi }{ 6 } \)
- (b)
\(\cfrac { \pi }{ 3 } \)
- (c)
\(\cfrac { \pi }{ 2 } \)
- (d)
\(\cfrac { 5\pi }{ 2 } \)
Let \(\vec { a } =\hat { i } +\hat { j } +\hat { k } .\vec { b } =\hat { i } -\hat { j } +2\hat { k } \) and \(\vec { c } =x\hat { i } +\left( x-2 \right) \hat { j } -\hat { k } \)the vector \(\vec { c } \) lies is in the plane of \(\vec { a } \) and \(\vec { b } \) , then x equals
- (a)
0
- (b)
1
- (c)
-4
- (d)
-2
If the vector \(\hat { i } -2\hat { j+ } 3\hat { k } ,-2\hat { i } +3\hat { j } -4\hat { k } ,\lambda \hat { i } -\hat { j } +2\hat { k } \) are coplanar, then the value of \(\lambda \) is equal to
- (a)
0
- (b)
1
- (c)
2
- (d)
3
Find \(\lambda \) if the vectors \(\hat { i } -\hat { j } +\hat { k } ,3\hat { i } +\hat { j } +2\hat { k } \) and \(\hat { i } -\lambda \hat { j } -3\hat { k } \) are coplanar
- (a)
5
- (b)
12
- (c)
15
- (d)
8
If \(\left| \vec { a } -\vec { b } \right| =\left| \vec { a } \right| =\left| \vec { b } \right| =1\) then the angle between \(\vec { a } \quad and\quad \quad \vec { b } \) is
- (a)
\(\cfrac { \pi }{ 3 } \)
- (b)
\(\cfrac { 3\pi }{ 4 } \)
- (c)
\(\cfrac { \pi }{ 2 } \)
- (d)
00
\(\left( \vec { a } .\hat { i } \right) ^{ 2 }+\left( \vec { a } .\hat { j } \right) ^{ 2 }+\left( \vec { a } .\hat { k } \right) ^{ 2 }\) is equal to
- (a)
1
- (b)
\(\vec { a } \)
- (c)
\(-\vec { a } \)
- (d)
\(\left| \vec { a } \right| ^{ 2 }\)
If \(\vec { a } =\hat { i } +2\hat { j } +\hat { k } ,\vec { b } =\hat { i } -\hat { j } +\hat { k } ,\hat { c } =\hat { i } +\hat { j } -\hat { k } \) . A vector coplanar to \(\vec { a } and\quad \vec { b } \) has a projection along \(\vec { c } \) of magnitude \(\cfrac { 1 }{ \sqrt { 3 } } \), then the vector is
- (a)
\(4\hat { i } -\hat { j } +4\hat { k } \)
- (b)
\(4\hat { i } -\hat { j } -4\hat { k } \)
- (c)
\(2\hat { i } +\hat { j } +\hat { k } \)
- (d)
None of these