Three vectors \(\overrightarrow{a},\overrightarrow{b}\)and \(\overrightarrow{c}\) are such that \(|\overrightarrow{a}|=2,|\overrightarrow{b}|=3,|\overrightarrow{c}|=4,\) and \(\overrightarrow{a}+\overrightarrow{b}+\overrightarrow{c}=\overrightarrow{0}\) .Find \(4\overrightarrow{a}.\overrightarrow{b}+​​3\overrightarrow{b}.\overrightarrow{c}+3\overrightarrow{c}.\overrightarrow{a}.\)

Prove that the smallar angle between any two diagonals of a cube is cos^-1 \(({1\over3})\).

Let A, Band C represent the angles of a \(\triangle\)ABC and a, band c represent the lengths of the sides opposite to them, then prove that a² = b² + c² - 2bc cos A (Law of cosines)

Let \(\overrightarrow { a } =\hat { i } +\hat { j } +2\hat { k } \) and \(\overrightarrow { b } =\hat { i } +2\hat { j } +\hat { k } \) and \(\overrightarrow { c } \)  be a unit vectorin the plane determined by \(\overrightarrow { a } \) and \(\overrightarrow { b } \). If \(\overrightarrow { c } \) is perpendicular to the vector \(\hat { i } +\hat { j } +\hat { k } \) and makes an obtuse angle with \(\overrightarrow { a } \), then prove that \(\overrightarrow { c } =\frac { \hat { j } -\hat { k } }{ \sqrt { 2 } } \)

Let A, Band C represent the angles of a \(\triangle\)ABC and a, b, c represent the lengths of the sides opposite to them, then prove that a = b cos C + c cos B (Projection formula)

Let \(\vec{a}=2\hat{i}+\hat{j}-2\hat{k}\) and \(\vec{b}=\hat{i}+\hat{j}\) . Let \(\vec{c}\) be a vector such that \(\vec { a } .\vec { c } =\left| \vec { c } \right| ,\left| \vec { c } -\vec { a } \right| =2\sqrt { 2 } \) and the angle between  and  is 30^o.Then find the value of \(\left| (\vec { a } \times \vec { b } )\times \vec { c } \right| \)

If \(\vec{a}, \vec{b}\) and \(\vec{c}\) are three vectors such that \(\left| \vec { a } \right| =3.\left| \vec { b } \right| =4\) and \(\left| \vec { c } \right| =\sqrt { 24 } \) sum of any two vectors is orthogonal to the third vector, then find \(\left| \vec { a } +\vec { b } +\vec { c } \right| \).

If \(\left| \vec { a } \right| =\left| \vec { b } \right| =\left| \vec { a } +\vec { b } \right| \)=1 then prove that \({ \left| \vec { a } -\vec { b } \right| }=\sqrt { 3 } \)

Let \(\overrightarrow { a } ,\overrightarrow { b } \) and \(\overrightarrow { c } \) be unit vectors such that \(\overrightarrow { a } \) is perpendicular to both \(\overrightarrow { b } \) and \(\overrightarrow { c } \) and further the angle between \(\overrightarrow { b } \) and \(\overrightarrow { c } \) is \(\frac { \pi }{ 6 } \). Then \(\overrightarrow { a } =\pm 2\left( \overrightarrow { b } \times \overrightarrow { c } \right) \)

Let ABC be a triangle,\(\overrightarrow{BC}=\overrightarrow{a},\overrightarrow{CA}=\overrightarrow{b}\) and \(\overrightarrow{AB}=\overrightarrow{c}\). Then prove that \(\overrightarrow {a}\times \overrightarrow {b}=\overrightarrow {b}\times \overrightarrow {c}=\overrightarrow {c}\times \overrightarrow {a}.\)