Scalar triple product practice · Mixed difficulty

The volume of the tetrahedron whose vertices are the points $\hat{i}, \hat{i}+\hat{j}, \hat{i}+\hat{j}+\hat{k}$ and $2 \hat{i}+3 \hat{j}+\lambda \hat{k}$ is $1 / 6$ units. Then, the values of $\lambda$

If $\mathbf{a}=\frac{1}{\sqrt{10}}(3 \hat{\mathbf{i}}+\hat{\mathbf{k}})$ and $\mathbf{b}=\frac{1}{7}(2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}-6 \hat{\mathbf{k}})$, then the value of $(2-b) \cdot[(\mathbf{a} \times \mathbf{b}) \times(\mathbf{a}+2 \mathbf{b})]$ is

If ${\mathbf{\vec a}}$ and ${\mathbf{\vec b}}$ are unit vectors and $\left| {\mathbf{\vec a} + \mathbf{\vec b}} \right| = 1$, then $\left| {\mathbf{\vec a} + \mathbf{\vec b}} \right|$ is equal to

A parallelopiped is formed by planes drawn parallel to coordinate axes through the points $A=(1,2,3)$ and $B=(9,8,5)$. The volume of that parallelopiped is equal to (in cubic wits)

If $\vec{a}, \vec{b}, \vec{c}$ are any three vectors such that $(\vec{a}+\vec{b}) \cdot \vec{c}=(\vec{a}-\vec{b}) \cdot \vec{c}=0,$ then $(\vec{a} \times \vec{b}) \times \vec{c}$ is

If $\mathbf{r} \cdot \mathbf{i}=\mathbf{r} \cdot \mathbf{j}=\mathbf{r} \cdot \mathbf{k}$ and $|\mathbf{r}|=3$, then $\mathbf{r}=$

Let the vectors
$(2 + a + b)\widehat i + (a + 2b + c)\widehat j - (b + c)\widehat k$,
$(1 + b)\widehat i + 2b\widehat j - b\widehat k$ and $(2 + b)\widehat i + 2b\widehat j + (1 - b)\widehat k$, $a,b,c \in R$ be co-planar.
Then, which of the following is true?

The equation $|\mathbf{r}|^2-\mathbf{r} .(2 \mathbf{i}+4 \mathbf{j}-2 \mathbf{k})-10=0$ represents a

The length of the perpendicular from the origin to the plane passing through three non-collinear points a,b,c is

If ${a}, {b}, {c}$ are non-coplanar vectors and ${d}$ is a unit vector, then the value of $|({a} \cdot {d})({b} \times {c})+({b} \cdot {d})({c} \times {a})+({c} \cdot {d})({a} \times {b})|$

A unit vector coplanar with \widehat i + \widehat j + 2\hat k$\widehat \mathbf{i} + \widehat \mathbf{j} + 2\mathbf{\hat k}$ and \widehat i + 2\widehat j + \hat k$\widehat \mathbf{i} + 2\widehat \mathbf{j} + \mathbf{\hat k}$, and perpendicular to \widehat i + \widehat j + \hat k$\widehat \mathbf{i} + \widehat \mathbf{j} + \mathbf{\hat k}$ is

Let $\mathbf{\vec a},\;\mathbf{\vec b}\;\mathrm{and\;}\mathbf{\vec c}$ be three non-coplanar vectors, and let $\mathbf{\vec p},\;\mathbf{\vec q}\;\mathrm{and\;}\mathbf{\vec r}$ be vectors defined by
the relations

$\mathbf{\vec p} = \frac{{\mathbf{\vec b} \times \mathbf{\vec c}}}{{\left[ {\mathbf{\vec a}\;\mathbf{\vec b}\;\mathbf{\vec c}} \right]}}\mathrm{\;},\mathrm{\;}\mathbf{\vec q} = \frac{{\mathbf{\vec c} \times \mathbf{\vec a}}}{{\left[ {\mathbf{\vec a}\;\mathbf{\vec b}\;\mathbf{\vec c}} \right]}}\mathrm{and\;\;}\mathbf{\vec r} = \frac{{\mathbf{\vec a} \times \mathbf{\vec b}}}{{\left[ {\mathbf{\vec a}\;\mathbf{\vec b}\;\mathbf{\vec c}} \right]}}\mathrm{\;\;}$

Then, the value of the expression

$\left( {\mathbf{\vec a} + \mathbf{\vec b}} \right)\cdot\mathbf{\vec p} + \left( {\mathbf{\vec b} + \;\mathbf{\vec c}} \right)\cdot\mathbf{\vec q} + \left( {\mathbf{\vec c} + \mathbf{\vec a}} \right)\cdot\mathbf{\vec r}$ is equal to

Unit vectors $\vec{a}, \vec{b}, \vec{c}$ are coplanar. A unit vector $\vec{d}$ is perpendicular to them. If $(\vec{a} \times \vec{b}) \times(\vec{c} \times \vec{d})=\frac{1}{6} \hat{i}-\frac{1}{3} \hat{j}+\frac{1}{3} \hat{k}$ and the angle between $\vec{a}$ and $\vec{b}$ is $30^{\circ}$, then $\vec{c}$ is/are

If $\vec{a}, \vec{b}, \vec{c}$ are three non-coplanar vectors, then a vector $\vec{r}$ satisfying $\vec{r} \cdot \vec{a}=\vec{r} \cdot \vec{b}=\vec{r} \cdot \vec{c}=1$, is

If three conterminous edges of a parallelopiped are represented by $\mathbf{a}-\mathbf{b}, \mathbf{b}-\mathbf{c}$ and $\mathbf{c}-\mathbf{a}$, then its volume is

Statement 1: The vectors $\vec{a}, \vec{b}$ and $\vec{c}$ lie in the same plane if and only if $\vec{a} \cdot(\vec{b} \times \vec{c})=0$

Statement 2: The vectors $\vec{u}$ and $\vec{v}$ are perpendicular if and only if $\vec{u} \cdot \vec{v}=0$ where $\vec{u} \times \vec{v}$ is a vector perpendicular to the plane of $\vec{u}$ and $\vec{v}$ .

If $\mathbf{\vec a} = 2\mathbf{\hat i} - 3\mathbf{\hat j} + 5\mathbf{\hat k}$, $\mathbf{\vec b} = 3\mathbf{\hat i} - 4\mathbf{\hat j} + 5\mathbf{\hat k}$ and $\mathbf{\vec c} = 5\mathbf{\hat i} - 3\mathbf{\hat j} - 2\mathbf{\hat k}$, then the volume of the parallelopiped with coterminous edges $\mathbf{\vec a} + \mathbf{\vec b},\mathbf{\vec b} + \mathbf{\vec c},\mathbf{\vec c} + \mathbf{\vec a}$ is

If $\vec{a}=\hat{i}+2 \hat{j}+\hat{k}, \vec{b}=3(\hat{i}-\hat{j}+\hat{k}), \vec{a} \cdot \vec{c}=3$
$\vec{a} \times \vec{c}=\vec{b}$ then $\vec{a} \cdot[(\vec{c} \times \vec{b})-\vec{b}-\vec{c}]=$

If $|\mathbf{a}|=3,|\mathbf{b}|=4,|\mathbf{c}|=5$ and $\mathbf{a}+\mathbf{b}+\mathbf{c}=0$, then the angle between a and $b$ is

The vectors $\vec a = x\hat i + (x + 1)\hat j + (x + 2)\hat k{\text{, }}$ $\vec{b}=(x+3) \hat{i}+(x+4) \hat{j}+(x+5) \hat{k}$
and, $\vec{c}=(x+6) \hat{i}+(x+7) \hat{j}+(x+8) \hat{k}$ are coplanar for

If \widehat i - \hat k, λ \widehat i + \widehat j + ( 1 - λ )\hat k$\widehat \mathbf{i} - \mathbf{\hat k},\;\mathrm{\lambda \;}\widehat \mathbf{i} + \widehat \mathbf{j} + \left( {1 - \lambda } \right)\mathbf{\hat k}$ and μ \widehat i + λ \widehat j + ( 1 + λ \widehat j - μ )\hat k$\mathrm{\mu }\;\widehat \mathbf{i} + \mathrm{\lambda \;}\widehat \mathbf{j} + \left( {1 + \mathrm{\lambda \;}\widehat \mathbf{j} - \mathrm{\mu }} \right)\mathbf{\hat k}$ are three coterminal edges of a parallelopiped, then its volume depends on

The value of $\lambda$, for which the four points $2\,\mathbf{\hat i} + 3\,\mathbf{\hat j} - \mathbf{\hat k},\,\;\mathbf{\hat i} + 2\,\mathbf{\hat j} + 3\,\mathbf{\hat k},\mathrm{\;}3\,\mathbf{\hat i} + 4\,\mathbf{\hat j} - 2\,\mathbf{\hat k},\mathrm{\;}\,\mathbf{\hat i} - \mathrm{\lambda }\,\mathbf{\hat j} + 6\,\mathbf{\hat k}$ are coplanar, is

The distance of the point $(-1,2,3)$ from the plane $\vec{r} \cdot(\hat{i}-2 \hat{j}+3 \hat{k})=10$ parallel to the line of the shortest distance between the lines $\vec{r}=(\hat{i}-\hat{j})+\lambda(2 \hat{i}+\hat{k})$ and $\vec{r}=(2 \hat{i}-\hat{j})+\mu(\hat{i}-\hat{j}+\hat{k})$ is

Let $O$ be the origin and the position vector of the point $P$ be $-\hat{i}-2 \hat{j}+3 k$. If the position vectors of the points $A, B$ and $C$ are $-2 \hat{i}+\hat{j}-3 k, 2 \hat{i}+4 \hat{j}-2 k$ and $-4 \hat{i}+2 \hat{j}-k$ respectively, then the projection of the vector $\overrightarrow{O P}$ on a vector perpendicular to the vectors $\overrightarrow{A B}$ and $\overrightarrow{A C}$ is

The value of $\vec{a} \cdot(\vec{b}+\vec{c}) \times(\vec{a}+\vec{b}+\vec{c})$, is

If the volume of a parallelepiped, whose coterminous edges are given by the vectors
$a = \hat i + \hat j + n\hat k,\,\,\,\,b = 2\hat i + 4\hat j - n\hat k$ and $c = \hat i + n\hat j + 3\hat k\left( {n \ge 0} \right),$ is 158 cu units, then

If the vectors $\vec{a}, \vec{b}, \vec{c}$ and $\vec{d}$ are coplanar vectors, then $(\vec{a} \times \vec{b}) \times(\vec{c} \times \vec{d})$ is equal to

Let r, a, b and c be four non-zero vectors such that $r.\,\,a = 0,$
$|r \times b| = |r||b|,\,\,|r \times c| = |r\parallel c\mid ,$ then [a b c] is

If $\alpha(\mathbf{a} \times \mathbf{b})+\beta(\mathbf{b} \times \mathbf{c})+\gamma(\mathbf{c} \times \mathbf{a})=0$, then

If $\vec{a}, \vec{b}, \vec{c}$ are vectors such that $[\vec{a} \vec{b} \vec{c}]=4$ then $[\vec{a} \times \vec{b} \vec{b} \times \vec{c} \vec{c} \times \vec{a}]=$