JEE Main Mathematics - Vector Algebra
Exam Duration: 60 Mins Total Questions : 30
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.
If \(\vec { a } \quad =\quad i+2j+3k,\quad and\quad \vec { b } \quad =\quad 3i+6j+2k,\) the vector in the direction of \(\vec { a } \) and having magnitude \(\left| \vec { b } \right| \) is
- (a)
7(i+2j+2k)
- (b)
\(\frac { 7 }{ 9 } (i+2j+2k)\)
- (c)
\(\frac { 7 }{ 3 } (i+2j+2k)\)
- (d)
NONE OF THESE
\(\vec { a } \times (\vec { b } \times \vec { c } )\) is equal to
- (a)
\((\vec { a } .\vec { c } )\vec { b } -(\vec { a } .\vec { b } )\vec { c } \)
- (b)
\((\vec { a } .\vec { b } )\vec { c }+(\vec { a } .\vec { c } )\vec { b } \)
- (c)
\((\vec { a } .\vec { c } )\vec { b }+(\vec { a } .\vec { b } )\vec { c } \)
- (d)
NONE OF THE ABOVE
A vector of unit length perpendicular to i+j and j+k is
- (a)
\(\frac { 1 }{ \sqrt { 3 } } (i+j+k)\)
- (b)
\(-\frac { 1 }{ \sqrt { 3 } } (i+j+k)\)
- (c)
\(\frac { 1 }{ \sqrt { 3 } } (i-j+k)\)
- (d)
\(\frac { 1 }{ \sqrt { 3 } } (i-j-k)\)
If \(\vec { X } .\vec { A } =0,\quad \vec { X } .\vec { B } =0,\quad \vec { X } .\vec { C } =0\) for some non-zero vector \(\vec X\), then value \([\vec { A } \vec { B } \vec { C } ]\) is
- (a)
0
- (b)
1
- (c)
2
- (d)
3
Let a,b,c be distinct non-negative numbers. If the vectors \(a\vec { i } +a\vec { j } +c\vec { k } ,\ \vec { i } +\vec { k },\ c\vec { i } +c\vec { j } +b\vec { k } \) lie in a plane, then c is
- (a)
the A.M. of a and b
- (b)
the G.M. of a and b
- (c)
the H.m. of a and b
- (d)
equal to zero
Let \(\vec { u } =\hat { i } +\hat { j } ,\quad \vec { v } =\hat { i } -\hat { j } \quad and\quad \vec { w } =\hat { i } +2\hat { j } +3\hat { k } \). If \(\hat { n } \) is a unit vector such that \(\vec { u } .\hat { n } =0\) and \(\vec { v } .\hat { n } =0\) then \(\left| \vec { w } .\hat { n } \right| \) is equal to
- (a)
0
- (b)
1
- (c)
2
- (d)
3
Three vectors \(a=\hat { i } +\hat { j } -\hat { k } ,b=-\hat { i } +2\hat { j } +\hat { k } and\quad c=-\hat { i } +2\hat { j } -\hat { k } ,\) then the unit vector perpendicular to both a+b and b+c is
- (a)
\(\frac { \hat { i } }{ \sqrt { 3 } } \)
- (b)
\(\hat { k } \)
- (c)
\(\frac { \hat { k } }{ \sqrt { 3 } } \)
- (d)
\(\frac { \hat { i } +\hat { j } +\hat { k } }{ \sqrt { 3 } } \)
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)
\(None\quad of\quad the\quad above\)
Let a,b and c be three non - zero vectors which are pairwise non - collinear. If a+3b is collinear with c and b + 2c is collinear with a, then a + 3b + 6c is
- (a)
a+c
- (b)
a
- (c)
c
- (d)
0
For any vector a, then the value of \((a\times \hat { i } )^{ 2 }+(a\times \hat { j } )^{ 2 }+(a\times \hat { k } )^{ 2 }\) is
- (a)
4a2
- (b)
2a2
- (c)
a2
- (d)
3a2
The vector \(\hat { i } +x\hat { j } +3\hat { k } \) is rotated through an angle \(\theta \) and doubled in magnitude, then it becomes \(4\hat { i } +(4x-2)\hat { j } +2\hat { k } .\) The values of x are
- (a)
\(\left\{ -\frac { 2 }{ 3 } ,2 \right\} \)
- (b)
\(\left( \frac { 1 }{ 3 } ,2 \right) \)
- (c)
\(\left( \frac { 2 }{ 3 } ,0 \right) \)
- (d)
\(\left\{ 2,7 \right\} \)
Let \(\vec { a } \),\(\vec { b } \),\(\vec { c } \) be three unit vectors such that 3 \(\vec { a } \) + 4 \(\vec { b } \) + 5 \(\vec { c } \) = 0. Then which of the following statements is true?
- (a)
\(\vec { a } is parallel to \vec b\)
- (b)
\(\vec { a } is perpendicular to \vec b\)
- (c)
\(\vec { a } is neither parallel nor perpendicular to \vec b\)
- (d)
none of the above
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 \((\vec { a } \times \vec { b } )^{ 2 }+(\vec { a } .\vec { b } )^{ 2 }=144\left| \vec { a } \right| =4\quad \left| \vec { b } \right| \)is Equal to
- (a)
16
- (b)
8
- (c)
3
- (d)
12
The vectors \(\overrightarrow { a } =x\hat { i } -2\hat { j } +5\hat { k } \) and \(\overrightarrow { b } =\hat { i } +y\hat { j } -z\hat { k } \) 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)
x = -1, y = 2, z = 5
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 O be the circumcentre G be the centroid and O' be the orthocentre of a \(\Delta ABC\) . Three vectors are taken through O and are represented by \(\vec a=\vec{OA},\vec b=\vec{OB}\) and \(\vec c=\vec{OC}\) then \(\vec a+\vec b+\vec c\) is
- (a)
\(\vec{OG}\)
- (b)
\(2\vec{OG}\)
- (c)
\(\vec{OO^\prime}\)
- (d)
none of these
\(\hat { x } \)and \(\hat { y } \) are two mutually perpendicular unit vectors , \(a\hat { x } +a\hat { y } +c(\hat { x } \times \hat { y } ),\quad \hat { x } +(\hat { x } \times \hat { y } )\quad c\hat { x } +c\hat { y } +b(\hat { x } \times \hat { y } )\) if the vectors and lie ina plaine then c is
- (a)
AM of a and b
- (b)
GM of a and B
- (c)
HM of a and b
- (d)
Equal to zero
Let \(\overrightarrow{a},\overrightarrow{b}\) and \(\overrightarrow{c}\) be the three non-zero and non-planar vectors and \(\overrightarrow{p},\overrightarrow{q}\) and \(\overrightarrow{r}\) be three vectors given by \(\overrightarrow{p}=\overrightarrow{a}+\overrightarrow{b}-2\overrightarrow{c},\overrightarrow{q}=3\overrightarrow{a}-2\overrightarrow{b}+\overrightarrow{c}\) and \(\overrightarrow{r}=\overrightarrow{a}-4\overrightarrow{b}+2\overrightarrow{c}\) If the volume of the parallelopiped determined by \(\overrightarrow{a},\overrightarrow{b}\) and \(\overrightarrow{c}\) V1 and that of the parallelopiped determined by \(\overrightarrow{p},\overrightarrow{q}\) and \(\overrightarrow{r}\) is V2 then V2 is equal to
- (a)
3 : 1
- (b)
7 : 1
- (c)
11 : 1
- (d)
15 : 1
The points with position vectors \(60\hat { i } +\hat { 3j } ,\quad 40\hat { i } -8\hat { j } \) and \(a\hat { i } +\hat { 52j } \) are colinear if
- (a)
a=-40
- (b)
a=40
- (c)
a=20
- (d)
None of these
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)
\(\cfrac { 5\pi }{ 2 } \)
Find the value of \(\lambda \) such that the vectors \(\vec { a } =2\hat { i } +\lambda \hat { j } +\hat { k } \) and \(\vec { b } =\hat { i } +2\hat { j } +3\hat { k } \) are orthogonal
- (a)
0
- (b)
1
- (c)
\(\cfrac { 3 }{ 2 } \)
- (d)
\(-\cfrac { 5 }{ 2 } \)
If the two volume of the parallelloppiped with conterminous edges \(-p\hat { j } ,5\hat { k } ,\hat { i } -\hat { j } +q\hat { k } \) and \(3\hat { i } -5\hat { j } \) is 8, then
- (a)
3pq+2=0
- (b)
3pq-2=0
- (c)
pq+2=0
- (d)
pq-2=0
Let \(\vec { a } ,\vec { b\quad } and\quad \vec { c } \) be vectors with magnitudes 3,4 5 respectiely and \(\vec { a } +\vec { b } +\vec { c } =0\), then the value of \(\vec { a } .\vec { b } +\vec { b } .\vec { c } +\vec { c } .\vec { a } \) is
- (a)
47
- (b)
25
- (c)
50
- (d)
-25
\((\hat i +\hat j) \times (\hat j +\hat k).(\hat k+\hat i)\) is equal to
- (a)
0
- (b)
1
- (c)
2
- (d)
none of these
if \(\vec { b } \) and \(\vec { c } \) are any two non - collinear unit vectors and \(\vec { a } \) is any vector, then \(\left( \vec { a } .\vec { b } \right) \vec { b } +\left( \vec { a } .\vec { c } \right) \vec { c } +\frac { \vec { a } .\left( \vec { b } \times \vec { c } \right) }{ \left| \vec { b } \times \vec { c } \right| } .\left( \vec { b } .\vec { c } \right) \) is equal to
- (a)
\(\vec { 0 } \)
- (b)
\(\vec { a } \)
- (c)
\(\vec { b } \)
- (d)
\(\vec { c } \)