If a vector \(\vec { \alpha } \) lies in the plane of \(\vec { \beta } \) and \(\vec { \gamma } \), then

If \(\vec { a } .\vec { b } =\vec { b } .\vec { c } =\vec { c } .\vec { a } =0\) , then the value of \([\vec { a } ,\vec { b } ,\vec { c } ]\) is

If \(\vec { a } ,\vec { b } ,\vec { c } \) are non-coplanar, non-zero vectors such that \([\vec { a } ,\vec { b } ,\vec { c } ]\) = 3, then \({ \{ [\vec { a } \times \vec { b } ,\vec { b } \times \vec { c } ,\vec { c } \times \vec { a } }]\} ^{ 2 }\) is equal to

If \(\vec { a } ,\vec { b } ,\vec { c } \) are three non-coplanar vectors such that \(\vec { a } \times (\vec { b } \times \vec { c } )=\frac { \vec { b } +\vec { c } }{ \sqrt { 2 } } \), then the angle between \(\vec { a } \ and \ \vec { b } \) is

If the volume of the parallelepiped with \(\vec { a } \times \vec { b } ,\vec { b } \times \vec { c } ,\vec { c } \times \vec { a } \)  as coterminous edges is 8 cubic units, then the volume of the parallelepiped with \((\vec { a } \times \vec { b } )\times (\vec { b } \times \vec { c } ),(\vec { b } \times \vec { c } )\times (\vec { c } \times \vec { a } )\) and \((\vec { c } \times \vec { a } )\times (\vec { a } \times \vec { b } )\)as coterminous edges is,

If \(\vec { a } \times (\vec { b } \times \vec { c } )=(\vec { a } \times \vec { b } )\times \vec { c } \) where \(\vec { a } ,\vec { b } ,\vec { c } \) are any three vectors such that \(\vec{b} \cdot \vec{c} \neq 0 \text { and } \vec{a} \cdot \vec{b} \neq 0\), then \(\vec { a } \) and \(\vec { c } \) are

If \(\vec { a } =2\hat { i } +3\hat { j } -\hat { k } ,\vec { b } =\hat { i } +2\hat { j } -5\hat { k } ,\vec { c } =3\hat { i } +5\hat { j } -\hat { k } ,\) then a vector perpendicular to \(\vec { a } \) and lies in the plane containing \(\vec { b } \) and \(\vec { c } \) is

The distance between the planes x + 2y + 3z + 7 = 0 and 2x + 4y + 6z + 7 = 0

The volume of the parallelepiped whose sides are given by \(\overset { \rightarrow }{ OA } =2\overset { \wedge }{ i } -3\overset { \wedge }{ j } \), \(\overset { \rightarrow }{ OB } =\overset { \wedge }{ i } +\overset { \wedge }{ j } -\overset { \wedge }{ k } \) and \(\overset { \rightarrow }{ OC } =3\overset { \wedge }{ i } -\overset { \wedge }{ k } \) is _____________

If  \(\left| \overset { \rightarrow }{ a } \right| =\left| \overset { \rightarrow }{ b } \right| =1\)such that \(\overset { \rightarrow }{ a } +2\overset { \rightarrow }{ b } \) and \(5\overset { \rightarrow }{ a } -\overset { \rightarrow }{ b } \) are perpendicular to each other, then the angle between \(\overset { \rightarrow }{ a } \) and \(\overset { \rightarrow }{ b } \) is _______________

If \(\lambda \overset { \wedge }{ i } +2\lambda \overset { \wedge }{ j } +2\lambda \overset { \wedge }{ k } \) is a unit vector, then the value of λ is _____________

If \(\left| \overset { \rightarrow }{ a } \times \overset { \rightarrow }{ b } \right| =\overset { \rightarrow }{ a } .\overset { \rightarrow }{ b } \), then the angle between the vector \(\overset { \rightarrow }{ a } \) and \(\overset { \rightarrow }{ b } \) is _____________

The value of \({ \left| \overset { \rightarrow }{ a } +\overset { \rightarrow }{ i } \right| }^{ 2 }\)+\({ \left| \overset { \rightarrow }{ a } +\overset { \rightarrow }{ j } \right| }^{ 2 }\)+\({ \left| \overset { \rightarrow }{ a } +\overset { \rightarrow }{ k } \right| }^{ 2 }\) if \(\left| a \right| =1\) is __________________

If \(\overset { \rightarrow }{ a } ,\overset { \rightarrow }{ b } ,\overset { \rightarrow }{ c } \) are three non - coplanar vectors, then \(\frac { \overset { \rightarrow }{ a } .\overset { \rightarrow }{ b } \times \overset { \rightarrow }{ c } }{ \overset { \rightarrow }{ c } \times \overset { \rightarrow }{ a } .\overset { \rightarrow }{ b } } +\frac { \overset { \rightarrow }{ b } .\overset { \rightarrow }{ a } \times \overset { \rightarrow }{ c } }{ \overset { \rightarrow }{ c } .\overset { \rightarrow }{ a } \times \overset { \rightarrow }{ b } } \) = _____________

Let \(\overset { \rightarrow }{ a } ,\overset { \rightarrow }{ b } ,\) and \(\overset { \rightarrow }{ c } \) be three vectors having magnitudes 1, 1, 2 respectively. If \(\overset { \rightarrow }{ a } \times \left( \overset { \rightarrow }{ a } \times \overset { \rightarrow }{ c } \right) +\overset { \rightarrow }{ b } =0\) then the acute angle between \(\overset { \rightarrow }{ a } \) and \(\overset { \rightarrow }{ c } \) is ___________ 

If \(\vec{ a } =\hat { -3i } -\hat { j } +\hat { 5k } \), \(\vec{b}=\hat{i}-\hat{2j}+\hat{k} \),  \(\vec{c}=\hat{4j}-\hat{5k} \ \) find\( \ {\vec a } .(\vec { b } \times \vec { c } )\)

If \(\vec { a } ,\vec { b } ,\vec { c } \) are three vectors, prove that \([\vec { a } +\vec { c } ,\vec { a } +\vec { b } ,\vec { a } +\vec { b } +\vec { c } ]\) = \([\vec { a } ,\vec { b } ,\vec { c } ]\)

The volume of the parallelepiped whose coterminus edges are \(7\hat { i } +\lambda \hat { j } -3\hat { k } ,\hat { i } +2\hat { j } -\hat { k } \), \(-3\hat { i } +7\hat { j } +5\hat { k } \) is 90 cubic units. Find the value of λ.

If \(\vec { a } =2\hat { i } +3\hat { j } -\hat { k } ,\vec { b } =3\hat { i } +5\hat { j } +2\hat { k } ,\vec { c } =-\hat { i } -2\hat { j } +3\hat { k } \), verify that
(i) \((\vec { a } \times \vec { b } )\times \vec { c } =(\vec { a } .\vec { c } )\times \vec { b } -(\vec { b } .\vec { c } )\vec { a } \)
(ii) \(\vec { a } \times (\vec { b } \times \vec { c } )=(\vec { a } .\vec { c } )\times \vec { b } -(\vec { a } .\vec { b } )\vec { c } \)

Find the vector and Cartesian equations of the plane passing through the point with position vector \(2\hat { i } +6\hat { j } +3\hat { k } \) and normal to the vector \(\hat { i } +3\hat { j } +5\hat { k } \)

A force of magnitude 6 units acting parallel to \(\overset { \wedge }{ 2i } -\overset { \wedge }{ 2j } +\overset { \wedge }{ k } \) displaces the point of application from (1, 2, 3) to (5, 3, 7). Find the work done.

Find the area of the triangle whose vertices  are A(3, -1, 2) B(1, -1, -3) and C(4, -3, 1)

Find the Cartesian equation of a line passing through the points A(2, -1, 3) and B(4, 2, 1)

Find the parametric form of vector equation of the plane passing through the point (1, -1, 2) having 2, 3, 3 as direction ratios of normal to the plane.

If the planes \({ \overset { \rightarrow }{ r } }.\left( \overset { \wedge }{ i } +2\overset { \wedge }{ j } +3\overset { \wedge }{ k } \right) =7\) and \({ \overset { \rightarrow }{ r } }.\left( \lambda \overset { \wedge }{ i } +2\overset { \wedge }{ j } -7\overset { \wedge }{ k } \right) =26\) are perpendicular. Find the value of λ.

With usual notations, in any triangle ABC, prove the following by vector method.
(i) a² = b² + c² − 2bc cos A
(ii) b² = c² + a² − 2ca cos B
(iii) c² = a² + b² − 2ab cos C

With usual notations, in any triangle ABC, prove the following by vector method.
(i) a = b cos C + c cos B
(ii) b = c cos A + a cos C
(iii) c = a cos B + b cos A

A particle is acted upon by the forces \((\hat { 3i } -\hat { 2j } +\hat { 2k } )\) and \((\hat { 2i } +\hat { j } -\hat { k } )\) is displaced from the point (1, 3, -1 ) to the point (4, -1, λ). If the work done by the forces is 16 units, find the value of λ.

Prove by vector method that the diagonals of a rhombus bisect each other at right angles.

If G is the centroid of a ΔABC, prove that (area of ΔGAB) = (area of ΔGBC) = (area of ΔGCA) = \(\frac{1}{3}\) (area of ΔABC)

Let \(\vec { a } ,\vec { b } ,\vec { c } \)  be three non-zero vectors such that \(\vec { c } \) is a unit vector perpendicular to both \(\vec { a } \) and \(\vec { b } \). If the angle between  \(\vec { a } \) and \(\vec { b } \) is \(​​\frac { \pi }{ 6 } \), show that \({ [\vec { a } ,\vec { b } ,\vec { c } ] }^{ 2 }\) = \(\frac { 1 }{ 4 } { \left| \vec { a } \right| }^{ 2 }{ \left| \vec { b } \right| }^{ 2 }\)

Find the direction cosines of the straight line passing through the points (5, 6, 7) and (7, 9, 13). Also, find the parametric form of vector equation and Cartesian equations of the straight line passing through two given points.

Find the non-parametric form of vector equation, and Cartesian equation of the plane passing through the point (0, 1, -5) and parallel to the straight lines \(\vec { r } =(\hat { i } +2\hat { j } -4\hat { k } )+s(\hat { i } +3\hat { j } +6\hat { k } )\) and \(\hat { r } =(\hat { i } -3\hat { j } +5\hat { k } )+t(\hat { i } +\hat { j } -\hat { k } )\)

Dot product of a vector with vector \(\overset { \wedge }{ 3i } -5\overset { \wedge }{ k } \), \(2\overset { \wedge }{ i } +7\overset { \wedge }{ j } \) and \(\overset { \wedge }{ i } +\overset { \wedge }{ j } +\overset { \wedge }{ k } \) are respectively -1, 6 and 5. Find the vector.

Show that the lines \(\frac { x-1 }{ 3 } =\frac { y+1 }{ 2 } =\frac { z-1 }{ 5 } \) and \(\frac { x+2 }{ 4 } =\frac { y-1 }{ 3 } =\frac { z+1 }{ -2 } \) do not intersect

By vector method, prove that cos(α + β) = cos α cos β -  sin α sin β

Prove by vector method that the perpendiculars (attitudes) from the vertices to the opposite sides of a triangle are concurrent.

Prove by vector method that sin(α + β ) = sin α cos β + cos α sin β

If \(\vec { a } =-2\hat { i } +3\hat { j } -2\hat { k } ,\vec { b } =3\hat { i } -\hat { j } +3\hat { k } ,\vec { c } =2\hat { i } -5\hat { j } +\hat { k } \) find \((\vec { a } \times \vec { b } )\times \vec { c } \) and \((\vec { a } \times \vec { b } )\times \vec { c } \). State whether they are equal.

Show that the lines \(\frac { x-2 }{ 1 } =\frac { y-3 }{ 1 } =\frac { z-4 }{ 3 } \) and \(\frac{x-1}{-3}=\frac{y-4}{2}=\frac{z-5}{1}\) coplanar. Also, find the plane containing these lines.

Show that the points A, B, C with position vector \(2\overset { \wedge }{ i } -\overset { \wedge }{ j } +\overset { \wedge }{ k } ,\overset { \wedge }{ i } -3\overset { \wedge }{ j } -5\overset { \wedge }{ k } \) and \(3\overset { \wedge }{ i } -4\overset { \wedge }{ j } +4\overset { \wedge }{ k } \) respectively are the vector of a right angled, triangle. Also, find the remaining angles of the triangle.

Find the vector and Cartesian equation of the plane passing through the point (1,1, -1) and perpendicular to the planes x + 2y + 3z - 7 = 0 and 2x - 3y + 4z = 0