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If $\mathbf{a}=2\mathbf{i}+4\mathbf{j}-5\mathbf{k}$ and $\mathbf{b}=\mathbf{i}+2\mathbf{j}+3\mathbf{k}$, then $|\mathbf{a}\times \mathbf{b}|$ is
The unit vector perpendicular to both $\mathbf{i}+\mathbf{j}$ and $\mathbf{j}+\mathbf{k}$ is
A unit vector in the plane of the vectors $2\mathbf{i}+\mathbf{j}+\mathbf{k},\,$ $\,\mathbf{i}-\mathbf{j}+\mathbf{k}$ and orthogonal to $5\mathbf{i}+2\mathbf{j}+6\mathbf{k}$ is
Let $\mathbf{a},\,\mathbf{b},\,\mathbf{c}$ be three vectors such that $\mathbf{a}\ne 0,$ and $\mathbf{a}\times \mathbf{b}=2\mathbf{a}\times \mathbf{c},\,\,|\mathbf{a}|\,=\,|\mathbf{c}|\,=\,1,\,|\mathbf{b}|\,=4$ and $|\mathbf{b}\times \mathbf{c}|\,=15.$ If $\mathbf{b}-2\mathbf{c}=\lambda \mathbf{a},$ then $\lambda$ equals to
The area of a triangle whose vertices are $A\,(1,\,-1,\,2),$ $B\,(2,\,1,\,-1)$ and $C\,(3,\,-1,\,2)$ is
If vertices of a triangle are $A(1,\,-1,\,2),\,B(2,\,0,\,-1)$ and $C(0,\,2,\,1),$ then the area of a triangle is
The area of triangle whose vertices are $(1,\,2,\,3),\,(2,\,5,\,-1)$ and $(-1,\,1,\,2)$ is
The area of a parallelogram whose two adjacent sides are represented by the vector $3\mathbf{i}-\mathbf{k}$ and $\mathbf{i}+2\mathbf{j}$ is
The area of the parallelogram whose diagonals are $\mathbf{a}=3\,\mathbf{i}+\mathbf{j}-2\mathbf{k}$ and $\mathbf{b}=\mathbf{i}-3\,\mathbf{j}+4\,\mathbf{k}$ is
The position vectors of the points A, B and C are $\mathbf{i}+\mathbf{j},\,\,\mathbf{j}+\mathbf{k}$ and $\mathbf{k}+\mathbf{i}$ respectively. The vector area of the $\Delta ABC=\pm \,\frac{1}{2}\overrightarrow{\alpha }$ where $\overrightarrow{\alpha }=$

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