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

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.

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 angle between the line \(\vec { r } =(2\hat { i } -\hat { j } +\hat { k } )+t(\hat { i } +2\hat { j } -2\hat { k } )\) and the plane \(\vec { r } =(6\hat { i } +3\hat { j } +2\hat { k } )=8\)

Find the acute angle between the following lines
2x = 3y = −z and 6x = − y = −4z.

If \(\overset { \rightarrow }{ a } =\overset { \wedge }{ i } +2\overset { \wedge }{ j } +3\overset { \wedge }{ k } \), \(\overset { \rightarrow }{ b } =-\overset { \wedge }{ i } +2\overset { \wedge }{ j } +\overset { \wedge }{ k } \) and \(\overset { \rightarrow }{ c } =3\overset { \wedge }{ i } +\overset { \wedge }{ j } \) find \(\frac { \lambda }{ c } \) such that \(\overset { \rightarrow }{ a } +\lambda \overset { \rightarrow }{ b } \) is perpendicular to \(\overset { \rightarrow }{ c } \)

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)

Forces \(2\overset { \wedge }{ i } +7\overset { \wedge }{ j } \), \(2\overset { \wedge }{ i } -5\overset { \wedge }{ j } +6\overset { \wedge }{ k } \), \(-\overset { \wedge }{ i } +2\overset { \wedge }{ j } -\overset { \wedge }{ k } \) act at a point P whose position vector is \(4\overset { \wedge }{ i } -3\overset { \wedge }{ j } -2\overset { \wedge }{ k } \). Find  the vector moment of the resultant of these forces acting at P about this point Q whose position vector is \(6\overset { \wedge }{ i } +\overset { \wedge }{ j } -3\overset { \wedge }{ k } \)

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 a line passing through a point (2, -1, 3) and parallel to line \({ \overset { \rightarrow }{ r } }=\left( \overset { \wedge }{ i } +\overset { \wedge }{ j } \right) +t\left( 2\overset { \wedge }{ i } +\overset { \wedge }{ j } -2\overset { \wedge }{ k } \right) \)

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.

Find the equation of the plane containing the line of intersection of the planes x + y + Z - 6 = 0 and 2x + 3y + 4z + 5 = 0 and passing through the point (1, 1, 1)

Let \(\overset { \rightarrow }{ a } ,\overset { \rightarrow }{ b } ,\overset { \rightarrow }{ c } \) be unit vectors such \(\overset { \rightarrow }{ a } .\overset { \rightarrow }{ b } =\overset { \rightarrow }{ a } .\overset { \rightarrow }{ c } =0\) and the angle between \(\overset { \rightarrow }{ b } \) and \(\overset { \rightarrow }{ c } \) is \(\frac { \pi }{ 6 } \). Prove that \(\overset { \rightarrow }{ a } =\pm 2\left( \overset { \rightarrow }{ b } \times \overset { \rightarrow }{ c } \right) \)