Let $\mathbf{a}=2 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}+\hat{\mathbf{k}}, \mathbf{b}=-\hat{\mathbf{j}}+\hat{\mathbf{k}}$. If $\mathbf{c}$ is a vector such that $\mathbf{a} \cdot \mathbf{c}=|\mathbf{c}|,|\mathbf{c}-\mathbf{a}|=2 \sqrt{2}$ and the angle between $\mathbf{a} \times \mathbf{b}$ and $\mathbf{c}$ is $\frac{\pi}{3}$, then $|(\mathbf{a} \times \mathbf{b}) \times \mathbf{c}|=$
If $\mathbf{a}, \mathbf{b}, \mathbf{c}$ are the position vectors of the points $A, B, C$ respectively, then match the items of list-I with those of list-II.
| $$ \text { List-I } $$ |
$$ \text { List-II } $$ |
||
|---|---|---|---|
| A. | $$ \text { } \begin{aligned} \mathbf{a} & =2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}, \\ \mathbf{b} & =3 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}+2 \hat{\mathbf{k}}, \\ \mathbf{c} & =4 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}} \end{aligned} $$ |
I. | $A, B, C$ are collinear |
| B. | $$ \text { } \begin{aligned} \mathbf{a} & =\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}, \\ \mathbf{b} & =3 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}+7 \hat{\mathbf{k}}, \\ \mathbf{c} & =-3 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}-5 \hat{\mathbf{k}} \end{aligned} $$ |
II. | $\triangle A B C$ is an isosceles triangle |
| C. | $$ \begin{aligned} &\text { }\\ &\begin{aligned} & a=2 \hat{i}-\hat{j}+\hat{k}, \\ & b=\hat{i}-3 \hat{j}-5 \hat{k}, \\ & c=-3 \hat{i}-4 \hat{j}-4 \hat{k} \end{aligned} \end{aligned} $$ |
III. | $\triangle A B C$ is a right-angled triangle |
| D. | $$ \begin{aligned} & a=\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \\ & b=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}, \\ & c=2 \hat{\mathbf{i}}-\hat{\mathbf{j}}+\hat{\mathbf{k}}, \end{aligned} $$ |
IV. | $\triangle A B C$ is a right-angled isosceles triangle |
| V. | $$ \triangle A B C \text { is an equilateral triangle } $$ |
||
$$ \text { The correct match is } $$
Let $\mathbf{a}=\hat{\mathbf{i}}+2 \hat{\mathbf{j}}-2 \hat{\mathbf{k}}$ and $\mathbf{b}=2 \hat{\mathbf{i}}-\hat{\mathbf{j}}-2 \hat{\mathbf{k}}$. If the orthogonal projection vector of $\mathbf{a}$ on $\mathbf{b}$ be $\mathbf{x}$ and the orthogonal projection vector of $\mathbf{b}$ on $\mathbf{a}$ be $\mathbf{y}$, then $|\mathbf{x}-\mathbf{y}|=$
Let $\mathbf{p}, \mathbf{q}, \mathbf{r}$ be three non-coplanar vectors and $\left[\begin{array}{lll}\mathbf{p} & \mathbf{q} & \mathbf{r}] \mathbf{a}=\mathbf{q} \times \mathbf{r},\left[\begin{array}{lll}\mathbf{p} & \mathbf{q} & \mathbf{r}] \mathbf{b}=\mathbf{r} \times \mathbf{p},[ \end{array}[\mathbf{p}\right. \\ \mathbf{q} & \mathbf{r}] \mathbf{c}=\mathbf{p} \times \mathbf{q} \text {. If }\end{array}\right. \mathbf{a}, \mathbf{b}, \mathbf{c}$ denote the coterminous edges of a parallelopiped, then its height with the base having a and $\mathbf{c}$ is
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