Vector Algebra

Let \(\overrightarrow a\), \(\overrightarrow b\) and \(\overrightarrow c\) be three vectors such that \(\left| {\overrightarrow a } \right| = \sqrt 3\),
$$\left| {\overrightarrow b } \right| = 5,\overrightarrow b .\overrightarrow c = 1...

Let the volume of a parallelopiped whose
coterminous edges are given by
\(\overrightarrow u = \widehat i + \widehat j + \lambda \widehat k\), \(\overrightarrow v = \widehat i + \widehat j + 3\widehat k\) and
$$\overrightarrow w = 2\wideh...

Let \(\overrightarrow a = \widehat i - 2\widehat j + \widehat k\) and \(\overrightarrow b = \widehat i - \widehat j + \widehat k\) be two
vectors. If \(\overrightarrow c\) is a vector such that $$\overrightarrow b \times \overrightarrow...

A vector \(\overrightarrow a = \alpha \widehat i + 2\widehat j + \beta \widehat k\left( {\alpha ,\beta \in R} \right)\) lies in the plane of the vectors, \(\overrightarrow b = \widehat i + \widehat j\) and $$\overrightarrow c = \wideha...

Let \(\overrightarrow a\)
, \(\overrightarrow b\)
and \(\overrightarrow c\)
be three unit vectors such that
\(\overrightarrow a + \vec b + \overrightarrow c = \overrightarrow 0\). If $$\lambda = \overrightarrow a .\vec b + \vec b.\...

If \(\overrightarrow a\)
and \(\overrightarrow b\)
are unit vectors, then the greatest value of
\(\sqrt 3 \left| {\overrightarrow a + \overrightarrow b } \right| + \left| {\overrightarrow a - \overrightarrow b } \right|\) is_____.

If \(\overrightarrow x\) and \(\overrightarrow y\) be two non-zero vectors such that
\(\left| {\overrightarrow x + \overrightarrow y } \right| = \left| {\overrightarrow x } \right|\) and $${2\overrightarrow x + \lambda \overrightarrow y...

If the volume of a parallelopiped, whose coterminus edges are given by the vectors \(\overrightarrow a = \widehat i + \widehat j + n\widehat k\), \(\overrightarrow b = 2\widehat i + 4\widehat j - n\widehat k\) and $$\overrightarrow c = ...

Let the vectors \(\overrightarrow a\), \(\overrightarrow b\), \(\overrightarrow c\)
be such that
\(\left| {\overrightarrow a } \right| = 2\), \(\left| {\overrightarrow b } \right| = 4\)
and \(\left| {\overrightarrow c } \right| = 4\). ...

Let x0 be the point of Local maxima of \(f(x) = \overrightarrow a .\left( {\overrightarrow b \times \overrightarrow c } \right)\), where \(\overrightarrow a = x\widehat i - 2\widehat j + 3\widehat k\), $$\overrightarrow b = - 2\widehat ...

If \(\overrightarrow a = 2\widehat i + \widehat j + 2\widehat k\), then the value of
$${\left| {\widehat i \times \left( {\overrightarrow a \times \widehat i} \right)} \right|^2} + {\left| {\widehat j \times \left( {\overrightarrow a \t...

The lines
\(\overrightarrow r = \left( {\widehat i - \widehat j} \right) + l\left( {2\widehat i + \widehat k} \right)\) and
$$\overrightarrow r = \left( {2\widehat i - \widehat j} \right) + m\left( {\widehat i + \widehat j + \widehat k} ...

Let a, b c \(\in\) R be such that a2
+ b2
+ c2
= 1. If \(a\cos \theta = b\cos \left( {\theta + {{2\pi } \over 3}} \right) = c\cos \left( {\theta + {{4\pi } \over 3}} \right)\),
where
\({\theta = {\pi \over 9}}\), then the angle be...

Let \(\overrightarrow a\), \(\overrightarrow b\) and \(\overrightarrow c\) be three unit vectors such that
\({\left| {\overrightarrow a - \overrightarrow b } \right|^2}\) + $${\left| {\overrightarrow a - \overrightarrow c } \right|^2}$...

Let the position vectors of points 'A' and 'B' be
\(\widehat i + \widehat j + \widehat k\) and \(2\widehat i + \widehat j + 3\widehat k\), respectively. A point
'P' divides the line segment AB internally in the
ratio
\(\lambda\) : 1 (
$$\l...

Let \(\overrightarrow a\) = \(\widehat i - \widehat j\), \(\overrightarrow b\) = \(\widehat i + \widehat j + \widehat k\) and \(\overrightarrow c\)
be a vector such that \(\overrightarrow a\) × \(\overrightarrow c\) + $$\overrightarro...

Let  \(\overrightarrow a = \widehat i + \widehat j + \sqrt 2 \widehat k,\)   \(\overrightarrow b = {b_1}\widehat i + {b_2}\widehat j + \sqrt 2 \widehat k\),    \(\overrightarrow c = 5\widehat i + \widehat j + \sqrt 2 \widehat k\)   be th...

Let \(\overrightarrow \alpha = 3\widehat i + \widehat j\) and \(\overrightarrow \beta = 2\widehat i - \widehat j + 3 \widehat k\)
. If \(\overrightarrow \beta = {\overrightarrow \beta _1} - \overrightarrow {{\beta _2}}\),
where $${\...

If a unit vector \(\overrightarrow a\) makes angles \(\pi\)/3 with \(\widehat i\) , \(\pi\)/ 4
with \(\widehat j\) and \(\theta\)\(\in\)(0, \(\pi\)) with \(\widehat k\), then a value of \(\theta\)
is :-

Let \(\mathop a\limits^ \to = 3\mathop i\limits^ \wedge + 2\mathop j\limits^ \wedge + x\mathop k\limits^ \wedge\) and \(\mathop b\limits^ \to = \mathop i\limits^ \wedge - \mathop j\limits^ \wedge + \mathop k\limits^ \wedge\)...

The sum of the distinct real values of \(\mu\), for which the vectors, \(\mu \widehat i + \widehat j + \widehat k,\)   \(\widehat i + \mu \widehat j + \widehat k,\)   \(\widehat i + \widehat j + \mu \widehat k\)  are co-planar, is :

Let \(\overrightarrow a\), \(\overrightarrow b\) and \(\overrightarrow c\) be three unit vectors, out of which vectors \(\overrightarrow b\) and \(\overrightarrow c\) are non-parallel. If \(\alpha\) and \(\beta\) are the angles whic...

Let \(\overrightarrow a = 3\widehat i + 2\widehat j + 2\widehat k\) and \(\overrightarrow b = \widehat i + 2\widehat j - 2\widehat k\) be two vectors. If a vector perpendicular to both the vectors
$$\overrightarrow a + \overrightarrow b ...

If the volume of parallelopiped formed by the vectors \(\widehat i + \lambda \widehat j + \widehat k\), \(\widehat j + \lambda \widehat k\) and \(\lambda \widehat i + \widehat k\) is minimum, then \(\lambda\) is
equal to :

Let \(\alpha\) \(\in\) R and the three vectors \(\overrightarrow a = \alpha \widehat i + \widehat j + 3\widehat k\), \(\overrightarrow b = 2\widehat i + \widehat j - \alpha \widehat k\) and $$\overrightarrow c = \alpha \widehat i - 2\...

Let  \(\overrightarrow a = \widehat i + 2\widehat j + 4\widehat k,\) \(\overrightarrow b = \widehat i + \lambda \widehat j + 4\widehat k\) and $$\overrightarrow c = 2\widehat i + 4\widehat j + \left( {{\lambda ^2} - 1} \right)\widehat k$...

Let \(\sqrt 3 \widehat i + \widehat j,\)    \(\widehat i + \sqrt 3 \widehat j\)  and   \(\beta \widehat i + \left( {1 - \beta } \right)\widehat j\) respectively be the position vectors of the points A, B and C with respect to the origin O. ...

Let \(\overrightarrow a = 2\widehat i + {\lambda _1}\widehat j + 3\widehat k,\,\,\)   \(\overrightarrow b = 4\widehat i + \left( {3 - {\lambda _2}} \right)\widehat j + 6\widehat k,\)  and  $$\overrightarrow c = 3\widehat i + 6\widehat j ...

If \(\overrightarrow \alpha\) = \(\left( {\lambda - 2} \right)\overrightarrow a + \overrightarrow b\)  and  \(\overrightarrow \beta = \left( {4\lambda - 2} \right)\overrightarrow a + 3\overrightarrow b\) be two given vectors $$\ov...

Let A (3, 0, –1), B(2, 10, 6) and C(1, 2, 1) be the vertices of a triangle and M be the midpoint of AC. If G
divides BM in the ratio, 2 : 1, then cos (\(\angle\)GOA) (O being the origin) is equal to :

The distance of the point having position vector \(- \widehat i + 2\widehat j + 6\widehat k\)
from the straight line passing through the point
(2, 3, – 4) and parallel to the vector, \(6\widehat i + 3\widehat j - 4\widehat k\) is :

Let \(\overrightarrow a = \widehat i + \widehat j + \widehat k,\overrightarrow c = \widehat j - \widehat k\) and a vector \(\overrightarrow b\) be such that \(\overrightarrow a \times \overrightarrow b = \overrightarrow c\) and $$\ove...

If \(\overrightarrow a ,\,\,\overrightarrow b ,\) and \(\overrightarrow C\) are unit vectors such that \(\overrightarrow a + 2\overrightarrow b + 2\overrightarrow c = \overrightarrow 0 ,\) then $$\left| {\overrightarrow a \times \over...

If the position vectors of the vertices A, B and C of a \(\Delta\) ABC are respectively \(4\widehat i + 7\widehat j + 8\widehat k,\)    \(2\widehat i + 3\widehat j + 4\widehat k,\) and \(2\widehat i + 5\widehat j + 7\widehat k,\) then the ...

Let \(\overrightarrow u\) be a vector coplanar with the vectors \(\overrightarrow a = 2\widehat i + 3\widehat j - \widehat k\) and \(\overrightarrow b = \widehat j + \widehat k\). If \(\overrightarrow u\) is perpendicular to $$\overrigh...

If the vector \(\overrightarrow b = 3\widehat j + 4\widehat k\) is written as the
sum of a vector \(\overrightarrow {{b_1}} ,\) paralel to \(\overrightarrow a = \widehat i + \widehat j\) and a vector \(\overrightarrow {{b_2}} ,\) perpe...

The area (in sq. units) of the parallelogram whose diagonals are along the vectors \(8\widehat i - 6\widehat j\) and \(3\widehat i + 4\widehat j - 12\widehat k,\) is :

Let \(\overrightarrow a = 2\widehat i + \widehat j -2 \widehat k\) and \(\overrightarrow b = \widehat i + \widehat j\).
Let \(\overrightarrow c\) be a vector such that \(\left| {\overrightarrow c - \overrightarrow a } \right| = 3\),
$$\...

In a triangle ABC, right angled at the vertex A, if the position vectors of A, B and C are respectively 3\(\widehat i\) + \(\widehat j\) \(-\) \(\widehat k\),   \(-\)\(\widehat i\) + 3\(\widehat j\) + p\(\widehat k\) and 5\(\widehat i\) + q...

Let ABC be a triangle whose circumcentre is at P. If the position vectors of A, B, C and P are \(\overrightarrow a ,\overrightarrow b ,\overrightarrow c\) and \({{\overrightarrow a + \overrightarrow b + \overrightarrow c } \over 4}\) ...

Let \(\overrightarrow a ,\overrightarrow b\) and \(\overrightarrow c\) be three unit vectors such that $$\overrightarrow a \times \left( {\overrightarrow b \times \overrightarrow c } \right) = {{\sqrt 3 } \over 2}\left( {\overrightarrow...

Let \(\overrightarrow a ,\overrightarrow b\) and \(\overrightarrow c\) be three non-zero vectors such that no two of them are collinear and $$\left( {\overrightarrow a \times \overrightarrow b } \right) \times \overrightarrow c = {1 \ov...

If $$\left[ {\overrightarrow a \times \overrightarrow b \,\,\,\,\overrightarrow b \times \overrightarrow c \,\,\,\,\overrightarrow c \times \overrightarrow a } \right] = \lambda {\left[ {\overrightarrow a\,\,\,\,\,\,\,\, \overrightarrow ...

If the vectors \(\overrightarrow {AB} = 3\widehat i + 4\widehat k\) and \(\overrightarrow {AC} = 5\widehat i - 2\widehat j + 4\widehat k\) are the sides of a triangle \(ABC,\) then the length of the median through \(A\) is :

Let \(\overrightarrow a\) and \(\overrightarrow b\) be two unit vectors. If the vectors \(\,\overrightarrow c = \widehat a + 2\widehat b\) and \(\overrightarrow d = 5\widehat a - 4\widehat b\) are perpendicular to each other, then the a...

Let \(ABCD\) be a parallelogram such that \(\overrightarrow {AB} = \overrightarrow q ,\overrightarrow {AD} = \overrightarrow p\) and \(\angle BAD\) be an acute angle. If \(\overrightarrow r\) is the vector that coincide with the altitud...

If \(\overrightarrow a = {1 \over {\sqrt {10} }}\left( {3\widehat i + \widehat k} \right)\) and \(\overrightarrow b = {1 \over 7}\left( {2\widehat i + 3\widehat j - 6\widehat k} \right),\) then the value
of $$\left( {2\overrightarrow a ...

Let \(\overrightarrow a\), \(\overrightarrow b\), \(\overrightarrow c\) be three non-zero vectors which are pairwise non-collinear. If $\overrightarrow a+3 \overrightarrow b$ is collinear with $\overrightarrow c$ and $\overrightarrow b+2...

The vectors \(\overrightarrow a\) and \(\overrightarrow b\) are not perpendicular and \(\overrightarrow c\) and \(\overrightarrow d\) are two vectors satisfying $$\overrightarrow b \times \overrightarrow c = \overrightarrow b \times ...

Let \(\overrightarrow a = \widehat j - \widehat k\) and \(\overrightarrow c = \widehat i - \widehat j - \widehat k.\) Then the vector \(\overrightarrow b\) satisfying $$\overrightarrow a \times \overrightarrow b + \overrightarrow c = ...