If \(\vec{a}\) and \(\vec{b}\) are parallel vectors, then \([\vec { a } ,\vec { c } ,\vec { b } ]\) is equal to

The volume of the parallelepiped with its edges represented by the vectors \(\hat { i } +\hat { j } ,\hat { i } +2\hat { j } ,\hat { i } +\hat { j } +\pi \hat { k } \) is

If \(\vec { a } \) and \(\vec { b } \) are unit vectors such that \([\vec { a } ,\vec { b },\vec { a } \times \vec { b } ]=\frac { 1}{ 4 } \), then the angle between \(\vec { a } \) and \(\vec { b } \) is

Consider the vectors \(\vec { a } ,\vec { b } ,\vec { c } ,\vec { d} \) such that \((\vec { a } \times \vec { b } )\times (\vec { c } \times \vec { d } )\) = \(\vec { 0 } \) Let \({ P }_{ 1 }\) and \({ P }_{ 2 }\) be the planes determined by the pairs of vectors \(\vec { a } ,\vec { b } \) and \(\vec { c } ,\vec { d } \) respectively. Then the angle between \({ P }_{ 1 }\) and \({ P }_{ 2 }\) 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

Let \(\overset { \rightarrow }{ a } \), \(\overset { \rightarrow }{ b } \)and \(\overset { \rightarrow }{ c } \)be three non- coplanar vectors and let  \(\overset { \rightarrow }{ p } ,\overset { \rightarrow }{ q } ,\overset { \rightarrow }{ r } \) be the vectors defined by the relations \(\overset { \rightarrow }{ P } =\frac { \overset { \rightarrow }{ b } \times \overset { \rightarrow }{ c } }{ \left[ \overset { \rightarrow }{ a } \overset { \rightarrow }{ b } \overset { \rightarrow }{ c } \right] } ,\overset { \rightarrow }{ q } =\frac { \overset { \rightarrow }{ c } \times \overset { \rightarrow }{ a } }{ \left[ \overset { \rightarrow }{ a } \overset { \rightarrow }{ b } \overset { \rightarrow }{ c } \right] } ,\overset { \rightarrow }{ r } =\frac { \overset { \rightarrow }{ a } \times \overset { \rightarrow }{ b } }{ \left[ \overset { \rightarrow }{ a } \overset { \rightarrow }{ b } \overset { \rightarrow }{ c } \right] } \) Then the value of \(\left( \overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } \right) .\overset { \rightarrow }{ p } +\left( \overset { \rightarrow }{ b } +\overset { \rightarrow }{ c } \right) .\overset { \rightarrow }{ q } +\left( \overset { \rightarrow }{ c } +\overset { \rightarrow }{ a } \right) .\overset { \rightarrow }{ r } \)= ____________

The number of vectors of unit length perpendicular to the vectors \(\left( \overset { \wedge }{ i } +\overset { \wedge }{ j } \right) \) and \(\left( \overset { \wedge }{ j } +\overset { \wedge }{ k } \right) \)is __________

If \(\overset { \rightarrow }{ d } =\overset { \rightarrow }{ a } \times \left( \overset { \rightarrow }{ b } \times \overset { \rightarrow }{ c } \right) +\overset { \rightarrow }{ b } \times \left( \overset { \rightarrow }{ c } \times \overset { \rightarrow }{ a } \right) +\overset { \rightarrow }{ c } \times \left( \overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } \right) \), then __________

If the vector \(\overset { \wedge }{ i } +\overset { \wedge }{ j } +\overset { \wedge }{ 2k } \), \(\overset { \wedge }{ -i } +\overset { \wedge }{ 2k } \) and \(2\overset { \wedge }{ i } +x\overset { \wedge }{ j } -y\overset { \wedge }{ k } \)  are mutually orthogonal, then the values of x, y, z are _________

If the work done by a force \(\overset { \rightarrow }{ F } =\overset { \wedge }{ i } +m\overset { \wedge }{ j } -\overset { \wedge }{ k } \) in moving the point of application from(1, 1, 1) to (3, 3, 3) along a straight line is 12 units, then m 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 _____________

For any three vectors \(\overset { \rightarrow }{ a } ,\overset { \rightarrow }{ b } \) and \(\overset { \rightarrow }{ c } \), \(\left( \overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } \right) .\left( \overset { \rightarrow }{ b } +\overset { \rightarrow }{ c } \right) \times \left( \overset { \rightarrow }{ c } +\overset { \rightarrow }{ a } \right) \) is _____________

Let \(\overset { \rightarrow }{ u } ,\overset { \rightarrow }{ v } ,\overset { \rightarrow }{ w } \) be vectors such that \(\overset { \rightarrow }{ u } +\overset { \rightarrow }{ v } +\overset { \rightarrow }{ w } =\overset { \rightarrow }{ 0 } \). If \(\left| \overset { \rightarrow }{ u } \right| \) = 3, \(\left| \overset { \rightarrow }{ v } \right| \) = 4, \(\left| \overset { \rightarrow }{ w } \right| \) = 5  then \(\overset { \rightarrow }{ u } .\overset { \rightarrow }{ v } +\overset { \rightarrow }{ v } .\overset { \rightarrow }{ w } +\overset { \rightarrow }{ w } .\overset { \rightarrow }{ u } \) is ______________

The length of the 丄^r from the origin to plane \(\overset { \rightarrow }{ r } .\left( \overset { \wedge }{ 3i } +4\overset { \wedge }{ j } +12\overset { \wedge }{ k } \right) \)= 26 is _____________

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 \(\hat { 2i } -\hat { j } +\hat { 3k } ,\hat { 3i } +\hat { 2j } +\hat { k } ,\hat { i } +\hat { mj } +\hat { 4k } \) are coplanar, find the value of m.

Find the volume of the parallelepiped whose coterminous edges are represented by the vectors \(-6\hat { i } +14\hat { j } +10\hat { k } ,14\hat { i } -10\hat { j } -6\hat { k } \) and \(2\hat { i } +4\hat { j } -2\hat { k } \)

Determine whether the three vectors \(2\hat { i } +3\hat { j } +\hat { k } \), \(\hat { i } -2\hat { j } +2\hat { k } \) and \(\hat { 3i } +\hat { j } +3\hat { k } \) are coplanar.

Prove that \((\vec { a } .(\vec { b } \times \vec { c } ))\vec { a } =(\vec { a } \times \vec { b } )\times (\vec { a } \times \vec { c } )\)

Find the parametric form of vector equation and Cartesian equations of the straight line passing through the point (−2, 3, 4) and parallel to the straight line \(\frac { x-1 }{ -4 } =\frac { y+3 }{ 5 } =\frac { 8-z }{ 6 } \)

Show that the points (2, 3, 4),(−1, 4, 5) and (8,1, 2) are collinear.

A variable plane moves in such a way that the sum of the reciprocals of its intercepts on the coordinate axes is a constant. Show that the plane passes through a fixed point

Find the direction cosines of the normal to the plane 12x + 3y − 4z = 65. Also, find the non-parametric form of vector equation of a plane and the length of the perpendicular to the plane from the origin.

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

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

If D is the midpoint of the side BC of a triangle ABC, then show by vector method that \({ \left| \vec { AB } \right| }^{ 2 }+{ \left| \vec { AC } \right| }^{ 2 }=2({ \left| \vec { AD} \right| }^{ 2 }+{ \left| \vec { BD } \right| }^{ 2 })\)

In triangle, ABC the points, D, E, F are the midpoints of the sides BC, CA and AB respectively. Using vector method, show that the area of ΔDEF is equal to \(\frac{1}{4}\)(area of ΔABC )

Prove by vector method that if a line is drawn from the centre of a circle to the midpoint of a chord, then the line is perpendicular to the chord.

Forces of magnit \(5\sqrt { 2 } \) and \(10\sqrt { 2 } \) units acting in the directions \(\hat { 3i } +\hat { 4j } +\hat { 5k } \) and \(\hat { 10i } +\hat { 6j } -\hat { 8k } \) respectively, act on a particle which is displaced from the point with position vector \(\hat { 4i } -\hat { 3j } -\hat { 2k } \) to the point with position vector \(\hat { 6i } +\hat { j } -\hat { 3k } \). Find the work done by the forces.

Find the torque of the resultant of the three forces represented by \(-\hat { 3i } +\hat { 6j } +\hat { 3k } \), \(\hat { 4i } -\hat { 10j } +\hat { 12k } \) and \(\hat { 4i } +\hat { 7j } \)  acting at the point with position vector \(\hat { 8i } -\hat { 6j } -\hat { 4k } \), about the point with position vector \(\hat { 18i } +\hat { 3j } -\hat { 9k } \)

Find the vector equation in parametric form and Cartesian equations of the line passing through (-4, 2, -3) and is parallel to the line  \(\frac { -x-2 }{ 4 } =\frac { y+3 }{ -2 } =\frac { 2z-6 }{ 3 } \)

Show that the straight lines x + 1=  2y = −12z and x = y + 2 = 6z − 6 are skew and hence find the shortest distance between them.

Prove that \(\left[ \overset { \rightarrow }{ a } +\overset { \rightarrow }{ b } +\overset { \rightarrow }{ c } ,\overset { \rightarrow }{ b } +\overset { \rightarrow }{ c } ,\overset { \rightarrow }{ c } \right] \)=\(\left[ \overset { \rightarrow }{ a } \overset { \rightarrow }{ b } \overset { \rightarrow }{ c } \right] \)

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

With usual notations, in any triangle ABC, prove by vector method that \(\frac { a }{ sinA } =\frac { b }{ sinB }=\frac { c }{ sinc }\)

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

Using vector method, prove that cos(α − β ) = cos α cos β +sin α sin β

Determine whether the pair of straight lines \(\vec { r } (2\hat { i } +\hat { 6j } +\hat { 3k } )+t(2\hat { i } +3\hat { j } +4\hat { k } )\), \(\vec { r } =(2\hat { j } -3\hat { k } )+s(\hat { i } +2\hat { j } +3\hat { k } )\) are parallel. Find the shortest distance between them.

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.