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Given $\mathbf{a}=\mathbf{i}+\mathbf{j}-\mathbf{k},\,\,\mathbf{b}=-\mathbf{i}+2\mathbf{j}+\mathbf{k}$ and $\mathbf{c}=-\mathbf{i}+2\mathbf{j}-\mathbf{k}.$ A unit vector perpendicular to both $\mathbf{a}+\mathbf{b}$ and $\mathbf{b}+\mathbf{c}$ is
The vectors $\mathbf{c},\,\,\,\mathbf{a}=x\mathbf{i}+y\mathbf{j}+z\mathbf{k}$ and $\mathbf{b}=\mathbf{j}$ are such that a, c, b form a right handed system, then c is
If A, B, C, D are any four points in space, then $|\overrightarrow{AB}\times \overrightarrow{CD}+\overrightarrow{BC}\times \overrightarrow{AD}+\overrightarrow{CA}\times \overrightarrow{BD}|$ is equal to (where $\Delta$ denotes the area of $\Delta ABC$)
If ${{(\mathbf{a}\times \mathbf{b})}^{2}}+{{(\mathbf{a}\,\,.\,\,\mathbf{b})}^{2}}=144$ and $|\mathbf{a}|\,=4,$ then $|\mathbf{b}|\,=$
$\mathbf{r}\times \mathbf{a}=\mathbf{b}\times \mathbf{a};\,\,\mathbf{r}\times \mathbf{b}=\mathbf{a}\times \mathbf{b};\,\,\mathbf{a}\ne 0;\,\,\mathbf{b}\ne 0;\,\,\mathbf{a}\ne \lambda \mathbf{b},\,\,$a is not perpendicular to b, then $\mathbf{r}=$
If $\mathbf{i},\,\,\mathbf{j},\,\,\mathbf{k}$ are unit orthonormal vectors and a is a vector, if $\mathbf{a}\times \mathbf{r}=\mathbf{j},$ then a . r is
A unit vector perpendicular to each of the vector $2\mathbf{i}-\mathbf{j}+\mathbf{k}$ and $3\mathbf{i}+4\mathbf{j}-\mathbf{k}$ is equal to
If $\overrightarrow{A}=3\mathbf{i}+\mathbf{j}+2\mathbf{k}$ and $\overrightarrow{B}=2\mathbf{i}-2\mathbf{j}+4\mathbf{k}$ and ? is the angle between $\overrightarrow{A}$ and $\overrightarrow{B},$ then the value of $\sin \theta $ is
A unit vector perpendicular to vector c and coplanar with vectors a and b is
$|\mathbf{a}\times \mathbf{b}{{|}^{2}}+\,{{(\mathbf{a}\,.\,\mathbf{b})}^{2}}=$

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