For any vector $r$,the expression $i \times(r \times i) + j \times(r \times j) + k \times(r \times k)$ is equal to:

Let $\vec{a}, \vec{b}$ and $\vec{c}$ be three vectors such that $\vec{a}=\vec{b} \times(\vec{b} \times \vec{c}) .$ If magnitudes of the vectors $\vec{a}, \vec{b}$ and $\vec{c}$ are $\sqrt{2}, 1$ and $2$ respectively and the angle between $\vec{b}$ and $\vec{c}$ is $\theta$ $(0 < \theta < \frac{\pi}{2})$,then the value of $1+\tan \theta$ is equal to:

Which of the following statements is true regarding the vector triple product $(a \times b) \times c$?

Let $\vec{a}=2 \hat{i}-\hat{j}+2 \hat{k}$ and $\vec{b}=\hat{i}+2 \hat{j}-\hat{k}$. Let a vector $\vec{v}$ be in the plane containing $\vec{a}$ and $\vec{b}$. If $\vec{v}$ is perpendicular to the vector $\vec{c}=3 \hat{i}+2 \hat{j}-\hat{k}$ and its projection on $\vec{a}$ is $19 \text{ units}$,then $|2 \vec{v}|^{2}$ is equal to .... .

Let $\vec{a}=-\hat{i}-\hat{j}+\hat{k}$,$\vec{a} \cdot \vec{b}=1$ and $\vec{a} \times \vec{b}=\hat{i}-\hat{j}$. Then $\vec{a}-6 \vec{b}$ is equal to

Let $\vec{a}=3 \hat{i}+\hat{j}$ and $\vec{b}=\hat{i}+2 \hat{j}+\hat{k}$. Let $\vec{c}$ be a vector satisfying $\vec{a} \times(\vec{b} \times \vec{c})=\vec{b}+\lambda \vec{c}$. If $\vec{b}$ and $\vec{c}$ are non-parallel,then the value of $\lambda$ is.