Vector Algebra

If \(\overrightarrow a = \left( {\widehat i + \widehat j + \widehat k} \right),\overrightarrow a .\overrightarrow b = 1\) and \(\overrightarrow a \times \overrightarrow b = \widehat j - \widehat k,\) then \(\overrightarrow b\) is

The unit vector which is orthogonal to the vector \(3\overrightarrow i + 2\overrightarrow j + 6\overrightarrow k\) and is coplanar with the vectors \(\,2\widehat i + \widehat j + \widehat k\) and $$\,\widehat i - \widehat j + \widehat k$...

If \(\overrightarrow a ,\overrightarrow b ,\overrightarrow c\) and \(\overrightarrow d\) are distinct vectors such that
\(\,\overrightarrow a \times \overrightarrow c = \overrightarrow b \times \overrightarrow d\) and $$\overrightarr...

The value of \('a'\) so that the volume of parallelopiped formed by \(\widehat i + a\widehat j + \widehat k,\widehat j + a\widehat k\) and \(a\widehat i + \widehat k\) becomes minimum is

If \(\overrightarrow u ,\overrightarrow v ,\overrightarrow w ,\) are three non-coplanar unit vectors and \(\alpha ,\beta ,\gamma\) are the angles between \(\overrightarrow u\) and \(\overrightarrow v\) and \(\overrightarrow w ,\) $$\ove...

Let \(\overrightarrow V = 2\overrightarrow i + \overrightarrow j - \overrightarrow k\) and \(\overrightarrow W = \overrightarrow i + 3\overrightarrow k .\) If \(\overrightarrow U\) is a unit vector, then the maximum value of the sca...

If \({\overrightarrow a }\) and \({\overrightarrow b }\) are two unit vectors such that \({\overrightarrow a + 2\overrightarrow b }\) and \({5\overrightarrow a - 4\overrightarrow b }\) are perpendicular to each other then the angle betwee...

Let \(V\) be the volume of the parallelopiped formed by the vectors \(\overrightarrow a = {a_1}\widehat i + {a_2}\widehat j + {a_3}\widehat k,\) \(\,\,\,\,\overrightarrow b = {b_1}\widehat i + {b_2}\widehat j + {b_3}\widehat k,\) $$\,\,...

If \(\overrightarrow a \,,\,\overrightarrow b\) and \(\overrightarrow c\) are unit vectors, then $${\left| {\overrightarrow a - \overrightarrow b } \right|^2} + {\left| {\overrightarrow b - \overrightarrow c } \right|^2} + {\left| {\ove...

Let \(\overrightarrow a = \overrightarrow i - \overrightarrow k ,\overrightarrow b = x\overrightarrow i + \overrightarrow j + \left( {1 - x} \right)\overrightarrow k\) and
$$\overrightarrow c = y\overrightarrow i - x\overrightarrow...

Let \(\overrightarrow A \left( t \right) = {f_1}\left( t \right)\widehat i + {f_2}\left( t \right)\widehat j\) and
$$$\overrightarrow B \left( t \right) = {g_1}\left( t \right)\overrightarrow i + {g_2}\left( t \right)\widehat j,t \in \lef...

Show, by vector methods, that the angular bisectors of a triangle are concurrent and find an expression for the position vector of the point of concurrency in terms of the position vectors of the vertices.

Find \(3-\)dimensional vectors \({\overrightarrow v _1},{\overrightarrow v _2},{\overrightarrow v _3}\) satisfying
$$\,{\overrightarrow v _1}.{\overrightarrow v _1} = 4,\,{\overrightarrow v _1}.{\overrightarrow v _2} = - 2,\,{\overrighta...

If \(\overrightarrow a \,,\,\overrightarrow b\) and \(\overrightarrow c\) are unit coplanar vectors, then the scalar triple product $$\left[ {2\overrightarrow a - \overrightarrow b ,2\overrightarrow b - \overrightarrow c ,2\overrightarr...

Let the vectors \(\overrightarrow a ,\overrightarrow b ,\overrightarrow c\) and \(\overrightarrow d\) be such that
$$\left( {\overrightarrow a \times \overrightarrow b } \right) \times \left( {\overrightarrow c \times \overrightarrow d...

If the vectors \(\overrightarrow a ,\overrightarrow b\) and \(\overrightarrow c\) form the sides \(BC,\) \(CA\) and \(AB\) respectively of a triangle \(ABC,\) then

Let \(a\) and \(b\) two non-collinear unit vectors. If \(u = a - \left( {a\,.\,b} \right)\,b\) and \(v = a \times b,\) then \(\left| v \right|\) is

Let \(a=2i+j-2k\) and \(b=i+j.\) If \(c\) is a vector such that \(a.\) \(c = \left| c \right|,\left| {c - a} \right| = 2\sqrt 2\) and the angle between \(\left( {a \times b} \right)\) and \(c\) is \({30^ \circ },\) then $$\left| {\left( {...

Let \(a=2i+j+k, b=i+2j-k\) and a unit vector \(c\) be coplanar. If \(c\) is perpendicular to \(a,\) then \(c =\)

Let \(u\) and \(v\) be units vectors. If \(w\) is a vector such that \(w + \left( {w \times u} \right) = v,\) then prove that \(\left| {\left( {u \times v} \right) \cdot w} \right| \le 1/2\) and that the equality holds if and only if \(u\) ...

If \(a = i + j + k,\overrightarrow b = 4i + 3j + 4k\) and \(c = i + \alpha j + \beta k\) are linearly dependent vectors and \(\left| c \right| = \sqrt 3 ,\) then

For three vectors \(u,v,w\) which of the following expression is not equal to any of the remaining three?

For any two vectors \(u\) and \(v,\) prove that
(a) \({\left( {u\,.\,v} \right)^2} + {\left| {u \times v} \right|^2} = {\left| u \right|^2}{\left| v \right|^2}\) and
(b) $$\left( {1 + {{\left| u \right|}^2}} \right)\left( {1 + {{\left| v...

Which of the following expressions are meaningful?

Prove, by vector methods or otherwise, that the point of intersection of the diagonals of a trapezium lies on the line passing through the mid-points of the parallel sides. (You may assume that the trapezium is not a parallelogram.)

Let \(OA=a,\) \(OB=10a+2b\) and \(OC=b\) where \(O,A\) and \(C\) are non-collinear points. Let \(p\) denote the area of the quadrilateral \(OABC,\) and let \(q\) denote the area of the parallelogram with \(OA\) and \(OC\) as adjacent sides....

If \(A,B\) and \(C\) are vectors such that \(\left| B \right| = \left| C \right|.\) Prove that
\(\left[ {\left( {A + B} \right) \times \left( {A + C} \right)} \right] \times \left( {B \times C} \right)\left( {B + C} \right) = 0\,\,.\)

A nonzero vector \(\overrightarrow a\) is parallel to the line of intersection of the plane determined by the vectors \(\widehat i,\widehat i + \widehat j\) and the plane determined by the vectors $$\widehat i - \widehat j,\widehat i + \wi...

If \(\overrightarrow b \,\) and \(\overrightarrow c \,\) are two non-collinear unit vectors and \(\overrightarrow a \,\) is any vector, then $$\left( {\overrightarrow a .\overrightarrow b } \right)\overrightarrow b + \left( {\overrightarro...

Let \(\overrightarrow u ,\overrightarrow v\) and \(\overrightarrow w\) be vectors such that \(\overrightarrow u + \overrightarrow v + \overrightarrow w = 0.\) If $$\left| {\overrightarrow u } \right| = 3,\left| {\overrightarrow v } \ri...

If \(\overrightarrow a ,\overrightarrow b ,\overrightarrow c\) are non coplanar unit vectors such that $$\overrightarrow a \times \left( {\overrightarrow b \times \overrightarrow c } \right) = {{\left( {\overrightarrow b + \overrightarr...

If \(\overrightarrow a ,\) \(\overrightarrow b\) and \(\overrightarrow c\) are three non coplanar vectors, then
$$\left( {\overrightarrow a + \overrightarrow b + \overrightarrow c } \right).\left[ {\left( {\overrightarrow a + \overri...

Let \(\overrightarrow a = \widehat i - \widehat j,\overrightarrow b = \widehat j - \widehat k,\overrightarrow c = \widehat k - \widehat i.\) If \(\overrightarrow d\) is a unit vector such that $$\overrightarrow a .\overrightarrow d =...

The vector \(\,{1 \over 3}\left( {2\widehat i - 2\widehat j + \widehat k} \right)\) is

If the vectors \(\overrightarrow b ,\overrightarrow c ,\overrightarrow d ,\) are not coplanar, then prove that the vector
$$\left( {\overrightarrow a \times \overrightarrow b } \right) \times \left( {\overrightarrow c \times \overrightar...

Let \(a, b, c\) be distinct non-negative numbers. If the vectors \(a\widehat i + a\widehat j + c\widehat k,\widehat i + \widehat k\) and \(c\widehat i + c\widehat j + b\widehat k\) lie in a plane, then \(c\) is

Let \(\vec a = 2\hat i - \hat j + \hat k,\vec b = \hat i + 2\hat j - \hat k\) and \(\overrightarrow c = \widehat i + \widehat j - 2\widehat k - 2\widehat k\) be three vectors. A vector in the plane of \({\overrightarrow b }\) and $${\ove...

In a triangle \(ABC, D\) and \(E\) are points on \(BC\) and \(AC\) respectively, such that \(BD=2DC\) and \(AE=3EC.\) Let \(P\) be the point of intersection of \(AD\) and \(BE.\) Find \(BP/PE\) using vector methods.

A unit vector coplanar with \(\overrightarrow i + \overrightarrow j + 2\overrightarrow k\) and \(\overrightarrow i + 2\overrightarrow j + \overrightarrow k\) and perpendicular to $$\overrightarrow i + \overrightarrow j + \overrighta...

Determine the value of \('c'\) so that for all real \(x,\) the vector
\(cx\widehat i - 6\widehat j - 3\widehat k\) and \(x\widehat i + 2\widehat j + 2cx\widehat k\) make an obtuse angle with each other.

Given that \(\overrightarrow a = \left( {1,1,1} \right),\,\,\overrightarrow c = \left( {0,1, - 1} \right),\,\overrightarrow a .\overrightarrow b = 3\) and \(\overrightarrow a \times \overrightarrow b = \overrightarrow c ,\) then $$\ove...

Let \(\overrightarrow A = 2\overrightarrow i + \overrightarrow k ,\,\overrightarrow B = \overrightarrow i + \overrightarrow j + \overrightarrow k ,\) and $$\overrightarrow C = 4\overrightarrow i - 3\overrightarrow j + 7\overrightarr...

For any three vectors \({\overrightarrow a ,\,\overrightarrow b ,}\) and \({\overrightarrow c ,}\)
$$\left( {\overrightarrow a - \overrightarrow b } \right)\,.\,\left( {\overrightarrow b - \overrightarrow c } \right)\, \times \,\left( {\...

If vectors \(\overrightarrow A ,\overrightarrow B ,\overrightarrow C\) are coplanar, show that
$$$\left| {\matrix{
{} & {\overrightarrow {a.} } & {} & {\overrightarrow {b.} } & {} & {\overrightarrow {c.} } \cr
{\overrightarrow {a....

In a triangle \(OAB,E\) is the midpoint of \(BO\) and \(D\) is a point on \(AB\) such that \(AD:DB=2:1.\) If \(OD\) and \(AE\) intersect at \(P,\) determine the ratio \(OP:PD\) using vector methods.

The components of a vector \(\overrightarrow a\) along and perpendicular to a non-zero vector \(\overrightarrow b\) are ......and .....respectively.

Let \(OA\) \(CB\) be a parallelogram with \(O\) at the origin and \(OC\) a diagonal. Let \(D\) be the midpoint of \(OA.\) Using vector methods prove that \(BD\) and \(CO\) intersect in the same ratio. Determine this ratio.

Let \(\overrightarrow a ,\overrightarrow b ,\overrightarrow c ,\) be three non-coplanar vectors and \(\overrightarrow p ,\overrightarrow q ,\overrightarrow r,\) are vectors defined by the relations $$\overrightarrow p = {{\overrightarrow b...

If the vectors \(a\widehat i + \widehat j + \widehat k,\,\,\widehat i + b\widehat j + \widehat k\) and \(\widehat i + \widehat j + c\widehat k\)
\(\left( {a \ne b \ne c \ne 1} \right)\) are coplannar, then the value of $${1 \over {\left( ...

If \(A, B, C, D\) are any four points in space, prove that -
$$\left| {\overrightarrow {AB} \times \overrightarrow {CD} + \overrightarrow {BC} \times \overrightarrow {AD} + \overrightarrow {CA} \times \overrightarrow {BD} } \right| = ...