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

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

Prove by vector method that the area of the quadrilateral ABCD having diagonals AC and BD is \(\frac { 1 }{ 2 } \left| \vec { AC } \times \vec { BD } \right| \).

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 \(\vec { a } ,\vec { b } ,\vec { c } \)  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 \({ [\vec { a } ,\vec { b } ,\vec { c } ] }^{ 2 }\) = \(\frac { 1 }{ 4 } { \left| \vec { a } \right| }^{ 2 }{ \left| \vec { b } \right| }^{ 2 }\)

For any vector \(\vec { a } \), prove that \(\hat { i } \times (\vec { a } \times \hat { i } )+\hat { j } \times (\vec { a } \times \hat { j } )+\hat { k } \times \vec { a } \times \hat { k } =2\vec { a } \).

Prove that \([\vec { a } -\vec { b } ,\vec { b } -\vec { c } ,\vec { c } -\vec { a } ]\) = 0

If \(\vec { a } ,\vec { b } ,\vec { c } ,\vec { d } \) are coplanar vectors, then show that \((\vec { a } \times \vec { b } )\times (\vec { c } \times \vec { d } )=\vec { 0 } \).

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

Find the angle between the lines \(\vec { r } =(\hat { i } +2\hat { j } +4\hat { k } )+t(2\hat { i } +2\hat { j } +\hat { k } )\) and the straight line passing through the points (5, 1, 4) and (9, 2, 12)