Vector Algebra

The number of vectors of unit length perpendicular to vectors \(\overrightarrow a = \left( {1,1,0} \right)\) and \(\overrightarrow b = \left( {0,1,1} \right)\) is

Let \(b = 4\widehat i + 3\widehat j\) and \(\overrightarrow c\) be two vectors perpendicular to each other in the \(xy\)-plane. All vectors in the same plane having projecttions \(1\) and \(2\) along \(\overrightarrow b\) and $$\overright...

The position vectors of the points \(A, B, C\) and \(D\) are \(3\widehat i - 2\widehat j - \widehat k,\,2\widehat i + 3\widehat j - 4\widehat k,\, - \widehat i + \widehat j + 2\widehat k\) and $$4\widehat i + 5\widehat j + \lambda \widehat ...

Let \(\overrightarrow a = {a_1}i + {a_2}j + {a_3}k,\,\,\,\overrightarrow b = {b_1}i + {b_2}j + {b_3}k\) and \(\overrightarrow c = {c_1}i + {c_2}j + {c_3}k\) be three non-zero vectors such that \(\overrightarrow c\) is a unit vector perp...

If \(\overrightarrow A = \left( {1,1,1} \right),\,\,\overrightarrow C = \left( {0,1, - 1} \right)\) are given vectors, then a vector \(B\) satifying the equations \(\overrightarrow A \times \overrightarrow B = \overrightarrow {\,C}\) a...

If \(\overrightarrow A \overrightarrow {\,B} \overrightarrow {\,C}\) are three non-coplannar vectors, then -
$${{\overrightarrow A .\overrightarrow B \times \overrightarrow C } \over {\overrightarrow C \times \overrightarrow A .\overrig...

\(A, B, C\) and \(D,\) are four points in a plane with position vectors \(a, b, c\) and \(d\) respectively such that
$$$\left( {\overrightarrow a - \overrightarrow d } \right)\left( {\overrightarrow b - \overrightarrow c } \right) = \left...

The points with position vectors \(a+b,\) \(a-b,\) and \(a+kb\) are collinear for all real values of \(k.\)

If \(X.A=0, X.B=0, X.C=0\) for some non-zero vector \(X,\) then \(\left[ {A\,B\,C} \right] = 0\)

\({A_1},{A_2},.................{A_n}\) are the vertices of a regular plane polygon with \(n\) sides and \(O\) is its centre. Show that
$$\sum\limits_{i = 1}^{n - 1} {\left( {\overrightarrow {O{A_i}} \times {{\overrightarrow {OA} }_{i + 1}...

For non-zero vectors \({\overrightarrow a ,\,\overrightarrow b ,\overrightarrow c },\) $$\left| {\left( {\overrightarrow a \times \overrightarrow b } \right).\overrightarrow c } \right| = \left| {\overrightarrow a } \right|\left| {\overrig...

Find all values of \(\lambda\) such that \(x, y, z,\)\(\, \ne\)\((0,0,0)\) and
$$\left( {\overrightarrow i + \overrightarrow j + 3\overrightarrow k } \right)x + \left( {3\overrightarrow i - 3\overrightarrow j + \overrightarrow k } \r...

The scalar \(\overrightarrow A .\left( {\overrightarrow B + \overrightarrow C } \right) \times \left( {\overrightarrow A + \overrightarrow B + \overrightarrow C } \right)\) equals :

Let \(\overrightarrow A ,\overrightarrow B\) and \({\overrightarrow C }\) be unit vectors suppose that \(\overrightarrow A .\overrightarrow B = \overrightarrow A .\overrightarrow C = 0,\) and thatthe angle between $${\overrightarrow B }$...

Let \(\overrightarrow A ,\overrightarrow B ,\overrightarrow C\) be vectors of length \(3, 4, 5\) respectively. Let \(\overrightarrow A\) be perpendicular to \(\overrightarrow B + \overrightarrow C ,\overrightarrow B\) to $$\overrightarr...