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

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

The angle between the line \(\vec { r } =(\hat { i } +2\hat { j } -3\hat { k } )+t(2\hat { i } +\hat { j } -2\hat { k } )\) and the plane \(\vec { r } .(\hat { i } +\hat { j } )+4=0\) is

The coordinates of the point where the line \(\vec { r } =(6\hat { i } -\hat { j } -3\hat { k } )+t(-\hat { i } +4\hat { j } )\) meets the plane \(\vec { r } .(\hat { i } +\hat { j } -\hat { k } )\) = 3 are

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

If the distance of the point (1, 1, 1) from the origin is half of its distance from the plane x + y + z + k = 0, then the values of k are

If the length of the perpendicular from the origin to the plane 2x + 3y + λz =1, λ > 0 is \(\frac{1}{5}\), then the value of λ is

(1) displacement
(2) length
(3) weight
(4) velocity

\(\overset { \rightarrow }{ a } ,\overset { \rightarrow }{ b } \) and \(\overset { \rightarrow }{ c } \) are said to be coplanar if
(1) \(\left[ \overset { \rightarrow }{ a } ,\overset { \rightarrow }{ b } ,\overset { \rightarrow }{ c } \right] \)=0
(2) \(\overset { \rightarrow }{ a } ,\overset { \rightarrow }{ b } ,\overset { \rightarrow }{ c } \) lie on the same plane
(3) They are either parallel or intersecting
(4) Skew lines

The equation of the plane at a distance p from the origin and perpendicular to the unit normal vector \(\overset { \wedge }{ d } \) is 
(1) \(\overset { \rightarrow }{ r } .\overset { \rightarrow }{ d } =p\)
(2) \(\overset { \rightarrow }{ r } .\overset { \wedge }{ d } =p\)
(3) \(\overset { \rightarrow }{ r } .\overset { \rightarrow }{ d } =q\) where \(q=p\left| \overset { \rightarrow }{ d } \right| \)
(4) \(\overset { \rightarrow }{ r } .\frac { \overset { \rightarrow }{ d } }{ \left| d \right| } =p\)

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

Show that the vectors \(\hat { i } +\hat { 2j } -\hat { 3k } \), \(\hat { 2i } -\hat { j } +\hat { 2k } \) and \(\hat { 3i } +\hat { j } -\hat { k } \)

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.

Find the non-parametric form of vector equation and Cartesian equations of the straight line passing through the point with position vector  \(4\hat { i } +3\hat { j } -7\hat { k } \) and parallel to the vector \(2\hat { i } -6\hat { j } +7\hat { k } \).

Find the vector and Cartesian equations of the plane passing through the point with position vector \(4\hat { i } +2\hat { j } -3\hat { k } \) and normal to vector \(2\hat { i } -\hat { j } +\hat { k } \)

Verify whether the line \(\frac { x-3 }{ -4 } =\frac { y-4 }{ -7 } =\frac { z+3 }{ 12 } \) lies in the plane 5x-y+z = 8.

Find the distance between the planes \(\vec { r } .(2\hat { i } -\hat { j } -2\hat { k } )\) = 6 and \(\vec { r } .(6\hat { i } -\hat { 3j } -\hat { 6k } )\) = 27

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

Find the magnitude and the direction cosines of the torque about the point (2, 0, -1) of a force \((\hat { 2i } +\hat { j } -\hat { k } )\), whose line of action passes through the origin

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.

Let \(\vec { a } =\hat { i } +\hat { j } +\hat { k } \),  \(\vec { b } =\hat { i } \)  and \(\vec { c } ={ c }_{ 1 }\hat { i } +{ c }_{ 2 }\hat { j } +{ c }_{ 3 }\hat { k } \). If \({ c }_{ 1 }=1\) and \({ c }_{ 2 }=2\), find \({ c }_{ 3 }\) such that \(\vec { a } ,\vec { b } \) and \(\vec { c } \) are coplanar.

\(\vec { a } =2\hat { i } +3\hat { j } -\hat { k } ,\vec { b } =-\hat { i } +2\hat { j } -4\hat { k } ,\vec { c } =\hat { i } +\hat { j } +\hat { k } \) then find the value of \((\vec { a } \times \vec { b } )\times (\vec { c } \times \vec { d } )\).

The vector equation in parametric form of a line is \(\vec { r } =(3\hat { i } -2\hat { j } +6\hat { k } )+t(2\hat { i } -\hat { j } +3\hat { k } )\). Find
(i) the direction cosines of the straight line
(ii) vector equation in non-parametric form of the line
(iii) Cartesian equations of the line.

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

If \(\vec { a } =\vec { i } -\vec { j } ,\vec { b } =\hat { i } -\hat { j } -4\hat { k } ,\vec { c } =3\hat { j } -\hat { k } \) and \(\vec { d } =2\hat { i } +5\hat { j } +\hat { k } \)
(i) \((\vec { a } \times \vec { b } )\times (\vec { c } \times \vec { d } )=[\vec { a } ,\vec { b } ,\vec { d } ]\vec { c } -[\vec { a } ,\vec { b } ,\vec { c } ]\vec { d } \)