Mathematics - Vector Algebra
Exam Duration: 45 Mins Total Questions : 30
The points with position vectors 7i-4j+7k, i-6j+10k, -i-3j+4k and 5i-j+5k form a
- (a)
square
- (b)
rhombus
- (c)
rectangle
- (d)
parallelogram but not a rhombus
If \(\vec { a } \quad =\quad i+2j+3k,\quad \vec { b } \quad =\quad -i+2j+k,\quad \vec { c } \quad =\quad 3i+j,\) then \(\vec { a } +t\vec { b } \) will be perpendicular to \(\vec { c } ,\) if t is equal to
- (a)
5
- (b)
4
- (c)
6
- (d)
2
\(\vec { a } ,\quad \vec { b } ,\quad \vec { c } ,\) 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)
\(\frac { 1 }{ 2 } \sqrt { 7 } \)
The value of s for which the points with position vectors -(j+k), 4i+5j+sk, 3i+9j+4k and 4(-i+j+k) are complanar is
- (a)
-1
- (b)
0
- (c)
1
- (d)
NONE OF THESE
Let \(\vec { a } \quad and\quad \vec { b } \) be two non-collinear units vectors. If \(\vec { u } =\vec { a } -(\vec { a } .\vec { b } )\vec { b } \quad and\quad \vec { v } =\vec { a } \times \vec { b } ,\quad then\quad \left| \vec { v } \right| \) equals
- (a)
\(\left| \vec { u } \right| \)
- (b)
\(\left| \vec { u } \right| +\left| \vec { v } .\vec { a } \right| \)
- (c)
\(2\left| \vec { v } \right| \)
- (d)
\(\left| \vec { u } \right| +\vec { u } .(\vec { a } +\vec { b } )\)
If a and b are unit vectors and \(\theta \) is the angle between them, then \(\left| \frac { a-b }{ 2 } \right| \) is
- (a)
\(sin\frac { \theta }{ 2 } \)
- (b)
\(cos\frac { \theta }{ 2 } \)
- (c)
\(tan\frac { \theta }{ 2 } \)
- (d)
\(sin\theta \)
If a and b are two vectors such that a.b<0 and |a.b| = |aXb|, then the angle between a and b is
- (a)
\(\frac { 3\pi }{ 4 } \)
- (b)
\(\frac { 2\pi }{ 3 } \)
- (c)
\(\frac { \pi }{ 4 } \)
- (d)
\(\frac { \pi }{ 3 } \)
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 non-coplanar and \(\vec { r } \) be any vector in space such that \(\vec { r } .\vec { a } =1,\)\(\vec { r } .\vec { b } =2\)and \(\vec { r } .\vec { c } =3\).if \(\left[ \vec { a } \vec { b } \vec { c } \right] =1\), then \(\vec { r } \) is equal to
- (a)
\(\vec { a } +2\vec { b } +3\vec { c } \)
- (b)
\(\vec { b } \times \vec { c } +2\vec { c } \times \vec { a } +3\vec { a } \times \vec { b } \)
- (c)
\(\left( \vec { b } .\vec { c } \right) \vec { a } +2\left( \vec { c } .\vec { a } \right) \vec { b } +3\left( \vec { a } .\vec { b } \right) \vec { c } \)
- (d)
none of the above
Vectors \(\vec a\) and \(\vec b\) are inclined at an angle \(\theta\) = 120o. If \(|\vec a|\) =1,\(|\vec b|\)=2 then {(\(\vec a\)+3\(\vec b\)) x (3\(\vec a\) -\(\vec b\) )}2 is equal to
- (a)
225
- (b)
275
- (c)
325
- (d)
300
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\(\vec { a } \)
The value of C so that for all real x, the vectors \(cx\hat { i } -6\hat { j } +3\hat { k } x\hat { i } +2\hat { j } +2cx\hat { k } \) make an obutse angle are
- (a)
c < 0
- (b)
0 < c < 4/3
- (c)
-4/3 < c < 0
- (d)
c > 0
The Projection of the vector \(2\hat { i } +3\hat { j } -2\hat { k } \) on the vector \(\hat { i } +2\hat { j } +3\hat { k } \)
- (a)
\(\frac { 1 }{ \sqrt { 14 } } \)
- (b)
\(\frac { 2 }{ \sqrt { 14 } } \)
- (c)
\(\frac { 3 }{ \sqrt { 14 } } \)
- (d)
none of these
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)
\(i\)
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
Let A be the given point whose position vector relative to an origin O be \(\vec { a } \) and \(\vec { ON } =\vec { n } \) Let \(\vec { r } \) be the position vector of any point P which lies on the plane and passing through A and perpendicular to ON. Then for any point P on the plane.
\(\vec { AP } .\vec { n } =0\)
\(\Rightarrow\) \(\left( \vec { r } -\vec { a } \right) .\vec { n } =0\)
\(\Rightarrow\) \(\vec { r } .\vec { n } =\vec { a } .\vec { n } \)
\(\Rightarrow\) \(\vec { r } .\hat { n } =p\)
Where P is perpendicular distance of the plane from origin .
The equation of the plane through the point \(\hat { i } +2\hat { j } -\hat { k } \) and perpendicular to the line of intersection of the planes \(\vec { r } .\left( 3\hat { i } -\hat { j } +\hat { k } \right) =1\) and \(\vec { r } .\left( \hat { i } +4\hat { j } -2\hat { k } \right) =2\) is
- (a)
\(\vec { r } .\left( 2\hat { i } +7\hat { j } -13\hat { k } \right) =29\)
- (b)
\(\vec { r } .\left( 2\hat { i } -7\hat { j } -13\hat { k } \right) =1\)
- (c)
\(\vec { r } .\left( 2\hat { i } -7\hat { j } +13\hat { k } \right) +25=0\)
- (d)
\(\vec { r } .\left( 2\hat { i } +7\hat { j } +13\hat { k } \right) =3\)
if are Two vectors of magnitude 2 incllined at an angle 60 then the angle between \(\vec { a } \) and \(\vec { a } +\vec { b } \) is
- (a)
30\(°\)
- (b)
60\(°\)
- (c)
45\(°\)
- (d)
None of these
If \(\vec { a } ,\vec { b } \) and \(\vec { c } \) are any three vectors, then \(\vec { a } \times \left( \vec { b } \times \vec { c } \right) =\left( \vec { a } \times \vec { b } \right) \times \vec { c } \) if and only if
- (a)
\(\vec { b } \) and \(\vec { c } \) are collinear
- (b)
\(\vec { a } \) and \(\vec { c } \) are collinear
- (c)
\(\vec { a } \) and \(\vec { b } \) are collinear
- (d)
none of these
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 \(\vec { a } \) and \(\vec { b } \) are two vectors making angles \(\theta \) with each other, then unit vectors along bisector of \(\vec { a } \) and \(\vec { b } \) is
- (a)
\(\pm \frac { \hat { a } +\hat { b } }{ 2 } \)
- (b)
\(\pm \frac { \hat { a } +\hat { b } }{ 2\cos { \theta } } \)
- (c)
\(\pm \frac { \hat { a } +\hat { b } }{ 2\cos { \theta /2 } } \)
- (d)
\(\pm \frac { \hat { a } +\hat { b } }{ \left| \hat { a } +\hat { b } \right| } \)
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
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)
\(- \frac {2}{3}\)
- (b)
\(\frac{1}{3}\)
- (c)
\(\frac{2}{3}\)
- (d)
2
If \(3\vec { a- } 5\vec { b } and\quad 2\vec { a } +\vec { b } \) are perpendicular to each other and \(\vec { a } +4\vec { b } ,-\vec { a } +\vec { b } \) are also mutually perpendicular then the cosine of the angle between and \(\vec { a } \) is \(\vec { b } \)
- (a)
\(\frac { 17 }{ 5\sqrt { 43 } } \)
- (b)
\(\frac { 19 }{ 5\sqrt { 43 } } \quad \)
- (c)
\(\frac { 21 }{ 5\sqrt { 43 } } \)
- (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)
\(\cfrac { 3 }{ 2 } \)
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 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
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