The angle between two vectors \(\vec{a}\) and \(\vec{b}\) with magnitudes\(\sqrt{3}\) and 4 respectively and \(\vec{a}.\vec{b}=2\sqrt{3}\) is ________ .

The direction cosines of the vector \(2\hat{i}+2\hat{j}-\hat{k}\) are ___________ .

The vector having initial and terminal points as (2, 5, 0) and (-3, 7, 4) respectively is _________ .

If \(|\overrightarrow { a } |=|\overrightarrow { b } |\) then

The value of the expression \({ \left| \overrightarrow { a } \times \overrightarrow { b } \right| }^{ 2 }+{ (\overrightarrow { a } .\overrightarrow { b } ) }^{ 2 }\) is ___________ .

The value of m for which the vectors \(3\hat { i } -6\hat { j } +\hat { k } \) and \(2\hat { i } -4\hat { j } +\lambda \hat { k } \) are parallel is __________ .

If \(\overrightarrow{a}=\hat{i}+2\hat{j}+2\hat{k},|\overrightarrow{b}|=5\) and the angle between \(\overrightarrow{a}\) and \(\overrightarrow{b}\) is \({\pi\over 6},\) then the area of the triangle formed by these two vectors as two sides, is

If \(\overrightarrow{a}=\hat{i}+\hat{j}+\hat{k},\overrightarrow{b}=2\hat{i}+x\hat{j}+\hat{k},\overrightarrow{c}=\hat{i}-\hat{j}+4\hat{k}\) and \(\overrightarrow{a}.(\overrightarrow{b}\times \overrightarrow{c})=70,\) then x is equal to

If the points whose position vectors \(10\hat{i}+3\hat{j},12\hat{i}-5\hat{j}\) and \(a\hat{i}+11\hat{j}\) are collinear then a is equal to

Vectors \(\overrightarrow{a}\) and \(\overrightarrow{b}\) are inclined at an angle \(\theta =120^o\). If \(|\overrightarrow{a}|=1,|\overrightarrow{b}|=2,\) then \([(\overrightarrow{a}+3\overrightarrow{b})\times (3\overrightarrow{a}-\overrightarrow{b})]^2\) is equal to