The values of ‘a’ for which the vectors are coplanar are:

For any vector $$\vec{a},$$​ the value of $$(\vec{a} \times \hat{i})^{2} + (\vec{a} \times \hat{j})^{2} + (\vec{a} \times \hat{k})^{2}$$​ is:

If $$S = \left\{ (x, y, z) \in \mathbb{E}^3 : x + y + z = 0 \right\}$$, then a Basis of S is​

If for any three non-null vectors ​​$$\vec{a}, \vec{b}, \vec{c}, \quad (\vec{a} \times \vec{b}) \times \vec{c} = \vec{a} \times (\vec{b} \times \vec{c})$$ then​

​$$\text{If } |\vec{\alpha}| = 10, |\vec{\beta}| = 1 \text{ and } \vec{\alpha} \cdot \vec{\beta} = 6, \text{ then value of } |\vec{\alpha} \times \vec{\beta}| \text{ is}$$​​

If $$W$$ be a proper subspace of a finite dimensional vector space, then​

If θ is the angle between any two vectors

​$$\vec{a} \text { and } \vec{b} \text {, then }|\vec{a} \times \vec{b}|=|\vec{a} \cdot \vec{b}| \text { when } \theta \text { is: }$$​​

​For two vectors ​

​$$\vec{A}=2 \widehat{i}+2 \widehat{j}+3 \widehat{k} \text { and }$$​​

​$$\vec{B}=5 \widehat{i}+2 \widehat{j}+7 \widehat{k}$$, find $$\vec{A} \cdot \vec{B}$$.​

For any vector $$\vec{a},$$​ the value of $$(\vec{a} \times \hat{i})^{2} + (\vec{a} \times \hat{j})^{2} + (\vec{a} \times \hat{k})^{2}$$​ is: