JEE Mathematics - Vector Algebra Chapter Sample Question Paper With Answer Key
Exam Duration: 60 Mins Total Questions : 50
If \(\left| \vec { \alpha } +\vec { \beta } \right| =\left| \vec { \alpha } -\vec { \beta } \right| \), then
- (a)
\(\vec { \alpha } \) is parallel to \(\beta\)
- (b)
\(\vec { \alpha } \) is perpendicular to `\(\vec { \beta } \)
- (c)
\(\left| \vec { \alpha } \right| =\left| \vec { \beta } \right| \)
- (d)
NONE OF THESE
The points with position vectors 60i+3j, 40i-8j and ai-52j are collinear, if
- (a)
a = -40
- (b)
a = 40
- (c)
a = 20
- (d)
NONE OF THESE
The two vectors \(\vec { a } \quad and\quad \vec { b } \) given as \(\vec { a } =2i+j+3k,\quad \vec { a } =4i-\lambda j+6k\) are parallel if
- (a)
\(\lambda \quad =\quad 2\)
- (b)
\(\lambda \quad =\quad -3\)
- (c)
\(\lambda \quad =\quad 3\)
- (d)
\(\lambda \quad =\quad -2\)
Let P, Q, R be three points with respective position vectors i+j, i-j and ai+bj+ck. The points P, Q, R are collinear, if
- (a)
a = b = c = 1
- (b)
a = b = c = 0
- (c)
a = 1, b and c arbitrary scalars
- (d)
a and b are arbitrary scalars and c = 0.
The position vectors of points A, B, C are i+j+k, i+2j+3k and 2i-j+k. The \(\triangle ABC\) is
- (a)
an isoceles triangle
- (b)
an equilateral triangle
- (c)
a scalene triangle
- (d)
a right angled triangle.
\(\vec { a } \quad =\quad i+2j+3k,\quad \vec { b } =2i+3j-2k;\) the projection of \(\vec { b } \quad on\quad \vec { a } \) is
- (a)
\(\frac { 1 }{ \sqrt { 14 } } \)
- (b)
\(\frac { 2 }{ \sqrt { 14 } } \)
- (c)
\(\frac { 3 }{ \sqrt { 14 } } \)
- (d)
NONE OF THESE
\(\vec { a } ,\quad \vec { b } ,\quad \vec { c } ,\) are three vectors such that \(\vec { a } +\vec { b } +\vec { c } =0\) and \(\left| \vec { a } \right| =3,\quad \left| \vec { b } \right| =5,\quad \left| \vec { c } \right| =7,\) then angle between \(\vec { a } \quad and\quad \vec { b } \) is,
- (a)
\({ 30 }^{ \circ }\)
- (b)
\({ 45}^{ \circ }\)
- (c)
\({ 60 }^{ \circ }\)
- (d)
NONE OF THESE
If \(\vec { a } \quad =\quad 3i-k,\quad \vec { b } \quad =\quad i+2j\) are adjacent sides of parallelogram, then its area is
- (a)
\(\frac { 1 }{ 2 } \sqrt { 17 } \)
- (b)
\(\frac { 1 }{ 2 } \sqrt { 14 } \)
- (c)
\(\sqrt { 41 } \)
- (d)
\(\frac { 1 }{ 2 } \sqrt { 7 } \)
A vector \(\vec { c } \) of magnitude \(5\sqrt { 6 } \) directed along the bisector of the angle between \(\vec { a } \quad =\quad 7i-4j-4k\) and \(\vec { b } \quad =\quad -2i-j+2k\) is
- (a)
\(\pm \frac { 5 }{ 3 } (2i+7j+k)\)
- (b)
\(\pm \frac { 3 }{ 5 } (i+7j+2k)\)
- (c)
\(\pm \frac { 5 }{ 3 } (i-2j+7k)\)
- (d)
\(\pm \frac { 5 }{ 3 } (i-7j+2k)\)
If the vector ai+j+k, i+bj+k and i+j+ck coplanar, the value if \(\frac { 1 }{ 1-a } +\frac { 1 }{ 1-b } +\frac { 1 }{ 1-c } ,\) is
- (a)
-1
- (b)
0
- (c)
1
- (d)
2
The scalar \(\vec { A } .(\vec { B } +\vec { C } )\times (\vec { A } +\vec { B } +\vec { C } )\) equals
- (a)
0
- (b)
\(2[\vec { A } ,\vec { B } ,\vec { C } ]\)
- (c)
\([\vec { A } ,\vec { B } ,\vec { C } ]\)
- (d)
NONE OF THESE
Forces 2i+7j, 2i-5j+6k, -i+2j-k act on a point P whose position vector is 4i-3j-2k. The magnitude of the moment of three forces acting at P about the point Q whose position vector is 6i+j-3k is
- (a)
\(\sqrt { 671 } \)
- (b)
\(\sqrt { 761 } \)
- (c)
\(\sqrt { 861 } \)
- (d)
NONE OF THESE
Which of the following expressions are meaningful?
- (a)
\(\vec { u } .(\vec { v } \times \vec { w } )\)
- (b)
\((\vec { u } .\vec { v } ).\vec { w } \)
- (c)
\((\vec { u } .\vec { v } )\times \vec { w } \)
- (d)
\(\vec { u } \times (\vec { v } .\vec { w } )\)
Let \(\vec { a } ,\vec { b } ,\vec { c } \) be three non-zero vectors, no two of which are collinear. If the vector \(\vec { a } +2\vec { b } \) is collinear with \(\vec { c } \) and \(\vec { b } +3\vec { c } \) is collinear with \(\vec { a } \), then \(\lambda\) being a non-zero scalar \(\vec { a } +2\vec { b } +6\vec { c } \) equals
- (a)
\(\lambda \vec { a } \)
- (b)
\(\lambda \vec { b } \)
- (c)
\(\lambda \vec { c } \)
- (d)
\( \vec { 0 } \)
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
If \(a=2\hat { i } +\hat { j } -\hat { k } ,b=\hat { j } +\hat { k } \) the vector c such that a.c = 4 and aXc = b is
- (a)
\(\hat { i } +\hat { j } +\hat { k } \)
- (b)
\(3\hat { i } +\hat { j } +\hat { k } \)
- (c)
\(\hat { i } +3\hat { j } +\hat { k } \)
- (d)
\(2\hat { i } -3\hat { j } -\hat { k } \)
Three points A,B and C with position vectors \({ a }_{ 1 }=3\hat { i } -2\hat { j } -\hat { k } ,{ a }_{ 2 }=\hat { i } +3\hat { j } +4\hat { k } and\quad { a }_{ 3 }=2\hat { i } +\hat { j } -2\hat { k } \) relative to an origin O. The distance of A from the plane OBC is (magnitude)
- (a)
5
- (b)
\(\sqrt { 3 } \)
- (c)
3
- (d)
\(2\sqrt { 3 } \)
If a,b,c are non - coplanar unit vectors such that \(a\times (b\times c)=\frac { b+c }{ \sqrt { 2 } } \) , then the angle between a and b is
- (a)
\(\frac { 3\pi }{ 4 } \)
- (b)
\(\frac { \pi }{ 4 } \)
- (c)
\(\frac { \pi }{ 2 } \)
- (d)
\(\pi \)
Let \(a=\hat { j } -\hat { k } \quad and\quad \quad c=\hat { i } -\hat { j } -\hat { k } ,\) then the vector b satisfying aXb+c=0 and a.b=3, is
- (a)
\(-\hat { i } +\hat { j } -2\hat { k } \)
- (b)
\(2\hat { i } -\hat { j } +2\hat { k } \)
- (c)
\(\hat { i } -\hat { j } -2\hat { k } \)
- (d)
\(\hat { i } +\hat { j } -2\hat { k } \)
Let \(a=\hat { i } +\hat { j } +\hat { k } ,b=\hat { i } -\hat { j } +2\hat { k } \quad and\quad c=x\hat { i } +(x-2)\hat { j } -\hat { k } .\) If the vector c lies in the plane of a and b, then x equals
- (a)
0
- (b)
1
- (c)
-4
- (d)
-2
Let u, v, w be such that |u| = 1, |v| =2, |w| =3 . If the projection v along u and v,w are perpendicular to each other, then |u - v + w| is equal to
- (a)
\(2\)
- (b)
\(\sqrt { 7 } \)
- (c)
\(\sqrt { 14 } \)
- (d)
\(14\)
Let\(\vec { a } \),\(\vec { b } \), and \(\vec { c } \) be mutually perpendicular vectors of the same magnitude. If a vector \(\vec { x } \) satisfies the equation \(\vec { a } \times \left[ \left( \vec { x } -\vec { b } \right) \times \vec { a } \right] +\vec { b } \times \left[ \left( \vec { x } -\vec { c } \right) \times \vec { b } \right] +\vec { c } \times \left[ \left( \vec { x } -\vec { a } \right) \times \vec { c } \right] =0\)then \(\vec { x } \) is given by
- (a)
\(\frac { 1 }{ 2 } \left( \vec { a } +\vec { b } +\vec { c } \right) \)
- (b)
\(\frac { 1 }{ 3 } \left(2 \vec { a } +\vec { b } +\vec { c } \right) \)
- (c)
\(\frac { 1 }{ 3 } \left( \vec { a } +\vec { b } +\vec { c } \right) \)
- (d)
\(\frac { 1 }{ 2 } \left( \vec { a } +\vec { b } -2\vec { c } \right) \)
A unit tangent vector at t = 2 on the curve x = t2 + 2, y = 4t - 5, z = 2t2 - 6t is
- (a)
\(\frac { 1 }{ \sqrt { 3 } } \left( \hat { i } +\hat { j } +\hat { k } \right) \)
- (b)
\(\frac { 1 }{ 3 } \left( 2\hat { i } +2\hat { j } +\hat { k } \right) \)
- (c)
\(\frac { 1 }{ \sqrt { 6 } } \left( 2\hat { i } +\hat { j } +\hat { k } \right) \)
- (d)
none of these
Let \(\vec { v } =2\hat { i } +\hat { j } -\hat { k } \) and \(\vec { w } =\hat { i } +3\hat { k } \). If \(\vec { u } \) is a unit vector, then maximum value of the scalar triple product \(\left[ \vec { u } \vec { v } \vec { w } \right] \)is
- (a)
-1
- (b)
\(\sqrt { 10 } +\sqrt { 6 } \)
- (c)
\(\sqrt { 59 } \)
- (d)
\(\sqrt { 60 }\)
If \(\left| \vec { a } +\vec { b } \right| =\left| \vec { a } -\vec { b } \right| \), then
- (a)
\(\vec a\) is parallel to \(\vec b\)
- (b)
\(\vec { a } \bot \vec { b } \)
- (c)
\(\left| \vec { a } \right| =\left| \vec { b } \right| \)
- (d)
none of these
The number of vectors of unit length perpendicular to the vectors \(\overrightarrow { a } \) = (1, 1, 0) and \(\overrightarrow { b } \)= (0, 1, 1) is
- (a)
1
- (b)
2
- (c)
3
- (d)
infinite
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 tetrahedron whose adjacent edges are represented by the vectors \(\vec {a}\), \(\vec {b}\) and \(\vec {c}\) is
- (a)
\(2\sqrt { \frac { 2 }{ 3 } } \)
- (b)
\(4\sqrt { \frac { 2 }{ 3 } } \)
- (c)
\(3\sqrt { \frac { 2 }{ 3 } } \)
- (d)
\(\sqrt { \frac { 2 }{ 3 } } \)
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 \(\vec { a } ,\quad \vec { b } \quad and\quad \vec { c } \) . If the system of vectors \(\vec { { e }^{ 1 } } ,\vec { { e }^{ 2 } } ,\vec { { e }^{ 3 } } \) is reciprocal to the system \(\vec { { e }_{ 1 } } ,\vec { { e }_{ 2 } } ,\vec { { e }_{ 3 } } ,\) then the value of \(\left( \vec { a } .\vec { { e }^{ 1 } } \right) \vec { { e }_{ 1 } } +\left( \vec { a } .\vec { { e }^{ 2 } } \right) \vec { { e }_{ 2 } } +\left( \vec { a } .\vec { { e }^{ 3 } } \right) \vec { { e }_{ 3 } } \)is
- (a)
0
- (b)
1
- (c)
\(\vec { a } \)
- (d)
3\(\vec { a } \)
If \(\vec { a } ,\vec { b, } \vec { c } \) are non-coplaner vectors then \(\frac { \vec { a } .(\vec { b, } \times \vec { c) } }{ \vec { b } (\vec { c } \times \vec { a } ) } +\frac { \vec { b } (\vec { c } \times \vec { a } ) }{ \vec { c } .(\vec { a } \times \vec { b } } +\frac { \vec { c } .(\vec { b } \times \vec { a) } }{ \vec { a } .(\vec { b, } \times \vec { c) } } \) is Equal to
- (a)
0
- (b)
1
- (c)
2
- (d)
None of these
if \(|\overrightarrow{a}|=4,|\overrightarrow{b}|=2\) and the angle between \(|\overrightarrow{a}|\) and \(|\overrightarrow{b}|\) is \(\frac{\pi}{6}\) then \((\overrightarrow{a}\times\overrightarrow{b})^{2}\) is
- (a)
48
- (b)
\((\overrightarrow{a})^{2}\)
- (c)
16
- (d)
32
If \(\overrightarrow{a},\overrightarrow{b},\overrightarrow{c}\) are non-coplanar non-zero vectors and \(\overrightarrow{r}\) is any vector in space then \(\left[ \overrightarrow { b } \overrightarrow { c } \overrightarrow { r } \right] \overrightarrow { a } +\left[ \overrightarrow { c } \overrightarrow { a } \overrightarrow { r } \right] \overrightarrow { b } +\left[ \overrightarrow { a } \overrightarrow { b } \overrightarrow { r } \right] \overrightarrow { c } \) is equal to
- (a)
\(3\left[ \overrightarrow { a } \overrightarrow { b } \overrightarrow { c } \right] \overrightarrow { r } \)
- (b)
\(\left[ \overrightarrow { a } \overrightarrow { b } \overrightarrow { c } \right] \overrightarrow { r } \)
- (c)
\(\left[ \overrightarrow { b } \overrightarrow { c } \overrightarrow { a } \right] \overrightarrow { r } \)
- (d)
\(\\ \left[ \overrightarrow { c } \overrightarrow { a } \overrightarrow { b } \right] \overrightarrow { r } \)
Let \(\vec { a } (x)=\sin { x } \hat { i } +(\cos { x) } \hat { j } \) and \(\vec { b } (x)=(\cos { 2x)\hat { i } +(sin2x)\hat { j } } \) be two variable vectors \((x\epsilon R)\) then \(\vec { a } (x)\quad \vec { b } (x)\) are
- (a)
collinear for unique Value of x
- (b)
perpendicular for infinntely many values of x
- (c)
Zero vectors for unique values of x
- (d)
none of the above
Let \(\vec { a } =2\hat { i } +\hat { j } -2\hat { k } \) and \(\vec { b } =\hat { i } +\hat { j } \) C is vector such that\(\vec { a } .\vec { c } =\left| c \right| ,\left| \vec { c } -\vec { a } \right| \) and the angle between \(\vec { a } \times \vec { b } and\vec { c } \) is 30\(°\) then \(|\vec { (a } \times \vec { b) } \vec { c } |\)
- (a)
\(\frac { 2 }{ 3 } \)
- (b)
\(\frac { 3 }{ 2 } \)
- (c)
2
- (d)
3
If \(\overrightarrow { a } =\hat { i } +\hat { j } +\hat { k } \) and \(\hat { b } =\hat { i } -\hat { j } ,\) then the vectors \(\left( \overrightarrow { a } .\hat { i } \right) \hat { i } +\left( \overrightarrow { a } .\hat { j } \right) +\hat { j } +\left( \overrightarrow { a } .\hat { k } \right) \hat { k } ,\left( \overrightarrow { b } .\hat { i } \right) \hat { i } +\left( \overrightarrow { b } .\hat { j } \right) \hat { j } +\left( \overrightarrow { b } .\hat { k } \right) \hat { k },\)and \(\hat { i } +\hat { j } -2\hat { k } \)
- (a)
are mutually perpendicular
- (b)
are coplanar
- (c)
from a parallelopiped of volume 6 units
- (d)
from a parallelopiped of volume 3 units
If \(\vec { a } ,\vec { b } ,\vec { c } \) be three vectors such that \(\vec { a } \times \vec { b } =\vec { c } \) and \(\vec { b } \times \vec { c } =\vec { a } \) then
- (a)
\(\vec { a } ,\vec { b } ,\vec { c } \) are orthogonal in pairs
- (b)
\(\left| \vec { a } \right| =\left| \vec { b } \right| =\left| \vec { c } \right| =1\)
- (c)
\(\left| \vec { a } \right| =\left| \vec { b } \right| =\left| \vec { c } \right| \neq 1\)
- (d)
\(\left| \vec { a } \right| \neq \left| \vec { b } \right| \neq \left| \vec { c } \right| \)
A vector \(\vec { a } \) has components 2p and 1 with respect to a rectangular cartesian system. This system is rotated through a certain angle about the origin in the clockwise sense. If with respect to new system \(\vec { a } \) has components p + 1 and 1, then
- (a)
p = 0
- (b)
p = 1 or p = -\(\frac{1}{3}\)
- (c)
p = -1 or p = \(\frac{1}{3}\)
- (d)
p = 1 or p = -1
If the vectors \(\vec c,\vec a=x\hat i+y\hat j+z\hat k\) and \(\vec b=\hat j\) are such that \(\vec a,\vec c\ and\ \vec b\) form a right handed system, then \(\vec c\) is
- (a)
\(z\hat i-x\hat k\)
- (b)
\(\vec 0\)
- (c)
\(y\hat j\)
- (d)
\(-z\hat i+x\hat k\)
If vectors \(\overrightarrow{a}\) and \(\overrightarrow{b}\) are non-collinear, then \(\frac { \overrightarrow { a } }{ |\overrightarrow { a } | } +\frac { \overrightarrow { b } }{ |\overrightarrow { b } | } \) is
- (a)
a unit vector
- (b)
in the plane of \(\overrightarrow{a}\) and \(\overrightarrow{b}\)
- (c)
equally inclined to \(\overrightarrow{a}\) and \(\overrightarrow{b}\)
- (d)
perpendicular to \(\overrightarrow{a}\) and \(\overrightarrow{b}\)
If \(\hat { i } \times \left( \vec { a } \times \hat { i } \right) \times \hat { j } \times \left( \vec { a } \times \hat { j } \right) \times \hat { k } \times \left( \vec { a } \times \hat { k } \right) =.........\left\{ \left( \vec { a } .\hat { i } \right) \hat { i } +\left( \vec { a } .\hat { j } \right) \hat { j } +\left( \vec { a } .\hat { k } \right) \hat { k } \right\} \)
- (a)
-1
- (b)
0
- (c)
2
- (d)
none of these
The position vectors of the points A,B,C are \(\left( 2\hat { i } +\hat { j } -\hat { k } \right) ,\left( 3\hat { i } -2\hat { j } +\hat { k } \right) \) and \(\left( \hat { i } +4\hat { j } -3\hat { k } \right) \) respectively. These position
- (a)
Form an isosceles triangle
- (b)
form a right angled triangle
- (c)
are coilnear
- (d)
form a scale triangle
The value of \(\lambda \) for which the vectors \(3\hat { i } -6\hat { j } +\hat { k } \) and \(2\hat { i } -4\hat { j } +\lambda \hat { k } \) are parallel is
- (a)
\(\cfrac { 2 }{ 3 } \)
- (b)
\(\cfrac { 3 }{ 2 } \)
- (c)
\(\cfrac { 5 }{ 2 } \)
- (d)
\(\cfrac { 2 }{ 5 } \)
Find the volume of the tetrahedron whose vertices are A(3,7,4), B(5,-2,3), C(-4,5,6) and D(1,2,3)
- (a)
12cu.units
- (b)
\(\cfrac { 23 }{ 3 } \) cu.units
- (c)
15 cu.units
- (d)
\(\cfrac { 46 }{ 3 } cu.units\)
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 \(\vec { a } ,\vec { b } ,\vec { c } \) are unit vectors, then \(\left| \vec { a } -\vec { b } \right| ^{ 2 }+\left| \vec { b } -\vec { c } \right| ^{ 2 }+\left| \vec { c } -\vec { a } \right| \) does not exceed
- (a)
4
- (b)
9
- (c)
8
- (d)
6
A unit vector perpendicular to te plane of \(\vec { a } =2\hat { i } -6\hat { j } -3\hat { k } \) and \(\vec { b } =4\hat { i } +3\hat { j } -\hat { k } \) is
- (a)
\(\cfrac { 4\hat { i } +3\hat { j } -\hat { k } }{ \sqrt { 26 } } \)
- (b)
\(\cfrac { 2\hat { i } -6\hat { j } -3\hat { k } }{ 7 } \)
- (c)
\(\cfrac { 3\hat { i } -2\hat { j } +6\hat { k } }{ 7 } \)
- (d)
\(\cfrac { 3\hat { 2i } -3\hat { j } -6\hat { k } }{ 7 } \)
\(\left| \vec { a } \times \vec { b } \right| ^{ 2 }+\left| \vec { a } .\vec { b } \right| ^{ 2 }\) =144 and \(\left| \vec { a } \right| =4\) and \(\left| \vec { b } \right| \) is equal to
- (a)
2
- (b)
6
- (c)
8
- (d)
20
If \(\left| \vec { a } \times \vec { b } \right| =4\) and \(\left| \vec { a } .\vec { b } \right| =2\) then \(\left| \vec { a } \right| ^{ 2 }\left| \vec { b } \right| ^{ 2 }\) is equal to
- (a)
2
- (b)
6
- (c)
8
- (d)
20
Find the value of \(\lambda \)so that the vectors \(2\hat { i } -4\hat { j } +\hat { k } \) and \(4\hat { i } -8\hat { j } +\lambda \hat { k } \) are parallel
- (a)
-1
- (b)
3
- (c)
-4
- (d)
2
If \(\vec { a } =2\hat { i } +\hat { j } +\hat { k } \), \(\vec { b } =\hat { i } -2\hat { j } -\hat { k } \), \(\vec { c } =\hat { i } +\hat { j } +\hat { k } \), then \(\vec { a } \times \left( \vec { b } \times \vec { c } \right) \)equals
- (a)
\(5\hat { i } -7\hat { j } -3\hat { k } \)
- (b)
\(5\hat { i } +7\hat { j } -3\hat { k } \)
- (c)
\(5\hat { i } -7\hat { j } +3\hat { k } \)
- (d)
zero vector
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