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

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.

Given three unit vectors $a, b, c$ such that $a \perp b$ and $a \parallel c$,then $a \times (b \times c)$ is

Let $\vec{v}$ be a vector such that $\vec{v} \times ((\hat{i}-\hat{k}) \times ((3\hat{i}+4\hat{j}) \times (\hat{j}+\hat{k}))) = \vec{0}$. Suppose $\vec{v} \cdot \hat{j} = -7$. Then $\vec{v} \cdot \hat{i}$ is

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 $a, b, c$ be three vectors. Determine the correctness of the following statements:
$(i)$ $(a \times b) \times c = (a \cdot c) b - (b \cdot c) a$
(ii) $a \times (b \times c) = (a \cdot c) b - (a \cdot b) c$