Vector Algebra

If the vectors \(\overrightarrow a = \widehat i - \widehat j + 2\widehat k,\,\,\,\,\,\overrightarrow b = 2\widehat i + 4\widehat j + \widehat k\,\,\,\) and \(\,\overrightarrow c = \lambda \widehat i + \widehat j + \mu \widehat k\) are m...

If \(\overrightarrow u ,\overrightarrow v ,\overrightarrow w\) are non-coplanar vectors and \(p,q\) are real numbers, then the equality $$\left[ {3\overrightarrow u \,\,p\overrightarrow v \,\,p\overrightarrow w } \right] - \left[ {p\overri...

The vector \(\overrightarrow a = \alpha \widehat i + 2\widehat j + \beta \widehat k\) lies in the plane of the vectors
\(\overrightarrow b = \widehat i + \widehat j\) and \(\overrightarrow c = \widehat j + \widehat k\) and bisects the ...

The non-zero vectors are \({\overrightarrow a ,\overrightarrow b }\) and \({\overrightarrow c }\) are related by \({\overrightarrow a = 8\overrightarrow b }\) and \({\overrightarrow c = - 7\overrightarrow b \,\,.}\) Then the angle betwee...

Let \(\overrightarrow a = \widehat i + \widehat j + \widehat k,\overrightarrow b = \widehat i - \widehat j + 2\widehat k\) and \(\overrightarrow c = x\widehat i + \left( {x - 2} \right)\widehat j - \widehat k\,\,.\) If the vectors $$\ov...

If \(\widehat u\) and \(\widehat v\) are unit vectors and \(\theta\) is the acute angle between them, then \(2\widehat u \times 3\widehat v\) is a unit vector for :

If \(\left( {\overrightarrow a \times \overrightarrow b } \right) \times \overrightarrow c = \overrightarrow a \times \left( {\overrightarrow b \times \overrightarrow c } \right)\) where \({\overrightarrow a ,\overrightarrow b }\) and $...

The values of a, for which the points \(A, B, C\) with position vectors \(2\widehat i - \widehat j + \widehat k,\,\,\widehat i - 3\widehat j - 5\widehat k\) and \(a\widehat i - 3\widehat j + \widehat k\) respectively are the vertices of a ...

If \(\overrightarrow a ,\overrightarrow b ,\overrightarrow c\) are non coplanar vectors and \(\lambda\) is a real number then $$\left[ {\lambda \left( {\overrightarrow a + \overrightarrow b } \right)\,\,\,\,\,\,\,\,{\lambda ^2}\overright...

If \(C\) is the mid point of \(AB\) and \(P\) is any point outside \(AB,\) then :

For any vector \({\overrightarrow a }\) , the value of \({\left( {\overrightarrow a \times \widehat i} \right)^2} + {\left( {\overrightarrow a \times \widehat j} \right)^2} + {\left( {\overrightarrow a \times \widehat k} \right)^2}\) is...

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

Let \(\overrightarrow a ,\overrightarrow b\) and \(\overrightarrow c\) be three non-zero vectors such that no two of these are collinear. If the vector \(\overrightarrow a + 2\overrightarrow b\) is collinear with \(\overrightarrow c\) ...

If \({\overrightarrow a ,\overrightarrow b ,\overrightarrow c }\) are non-coplanar vectors and \(\lambda\) is a real number, then the vectors $${\overrightarrow a + 2\overrightarrow b + 3\overrightarrow c ,\,\,\lambda \overrightarrow b ...

A particle acted on by constant forces \(4\widehat i + \widehat j - 3\widehat k\) and \(3\widehat i + \widehat j - \widehat k\) is displaced from the point \(\widehat i + 2\widehat j + 3\widehat k\) to the point $$\,5\widehat i + 4\widehat ...

Let \(\overrightarrow a ,\overrightarrow b\) and \(\overrightarrow c\) be non-zero vectors such that $$\left( {\overrightarrow a \times \overrightarrow b } \right) \times \overrightarrow c = {1 \over 3}\left| {\overrightarrow b } \right...

Let \(\overrightarrow u ,\overrightarrow v ,\overrightarrow w\) be such that \(\left| {\overrightarrow u } \right| = 1,\,\,\,\left| {\overrightarrow v } \right|2,\,\,\,\left| {\overrightarrow w } \right|3.\) If the projection $${\overright...

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

Let \(\overrightarrow u = \widehat i + \widehat j,\,\overrightarrow v = \widehat i - \widehat j\) and \(\overrightarrow w = \widehat i + 2\widehat j + 3\widehat k\,\,.\) If \(\widehat n\) is a unit vector such that $$\overrightarrow u .\...

If \(\left| {\matrix{ a & {{a^2}} & {1 + {a^3}} \cr b & {{b^2}} & {1 + {b^3}} \cr c & {{c^2}} & {1 + {c^3}} \cr } } \right| = 0\) and vectors \(\left( {1,a,{a^2}} \right),\,\,\)
\(\left( {1,b,{b^2}} \right)\) and $$\left( {...

Consider points \(A, B, C\) and \(D\) with position
vectors \(7\widehat i - 4\widehat j + 7\widehat k,\widehat i - 6\widehat j + 10\widehat k, - \widehat i - 3\widehat j + 4\widehat k\) and \(5\widehat i - \widehat j + 5\widehat k\) re...

A tetrahedron has vertices at \(O(0,0,0), A(1,2,1) B(2,1,3)\) and \(C(-1,1,2).\) Then the angle between the faces \(OAB\) and \(ABC\) will be :

\(\overrightarrow a \,,\overrightarrow b \,,\overrightarrow c\) are \(3\) vectors, such that \(\overrightarrow a + \overrightarrow b + \overrightarrow c = 0\) , $$\left| {\overrightarrow a } \right| = 1\,\,\,\left| {\overrightarrow b }...

If \(\overrightarrow a \times \overrightarrow b = \overrightarrow b \times \overrightarrow c = \overrightarrow c \times \overrightarrow a\) then \(\overrightarrow a + \overrightarrow b + \overrightarrow c =\)

If \(\overrightarrow u \,,\overrightarrow v\) and \(\overrightarrow w\) are three non-coplanar vectors, then $$\,\left( {\overrightarrow u + \overrightarrow v - \overrightarrow w } \right).\left( {\overrightarrow u - \overrightarrow v ...

If \(\overrightarrow a \,\,,\,\,\overrightarrow b \,\,,\,\,\overrightarrow c\) are vectors such that \(\left[ {\overrightarrow a \,\overrightarrow b \,\overrightarrow c } \right] = 4\) then $$\left[ {\overrightarrow a \, \times \overrighta...

If \(\left| {\overrightarrow a } \right| = 5,\left| {\overrightarrow b } \right| = 4,\left| {\overrightarrow c } \right| = 3\) thus what will be the value of $$\left| {\overrightarrow a .\overrightarrow b + \overrightarrow b .\overrightarr...

\(\overrightarrow a = 3\widehat i - 5\widehat j\) and \(\overrightarrow b = 6\widehat i + 3\widehat j\) are two vectors and \(\overrightarrow c\) is a vector such that \(\overrightarrow c = \overrightarrow a \times \overrightarrow b\)...

If the vectors $\overrightarrow{\mathbf{a}}, \overrightarrow{\mathbf{b}}$ and $\overrightarrow{\mathbf{c}}$ from the sides $B C, C A$ and $A B$ respectively of a triangle $A B C$, then :

If the vectors \(\overrightarrow c ,\overrightarrow a = x\widehat i + y\widehat j + z\widehat k\) and \(\widehat b = \widehat j\) are such that \(\overrightarrow a ,\overrightarrow c\) and \(\overrightarrow b\) form a right handed system...

If \(\left| {\overrightarrow a } \right| = 4,\left| {\overrightarrow b } \right| = 2\) and the angle between \({\overrightarrow a }\) and \({\overrightarrow b }\) is \(\pi /6\) then $${\left( {\overrightarrow a \times \overrightarrow b } \...