Questions in vectors-m

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If in a right angled triangle ABC, the hypotenuse $AB=p,$ then $\overrightarrow{AB}\,\,.\,\,\overrightarrow{AC}+\overrightarrow{BC}\,.\,\,\overrightarrow{BA}+\overrightarrow{CA}\,\,.\,\,\overrightarrow{CB}$ is equal to
A, B, C, D are any four points, then $\overrightarrow{AB}\,\,.\,\,\overrightarrow{CD}\,\,+\,\overrightarrow{\,BC}\,\,.\,\,\overrightarrow{AD}\,\,+\overrightarrow{CA}\,\,.\,\,\overrightarrow{BD}\,\,=$
The vector a coplanar with the vectors i and j, perpendicular to the vector $\mathbf{b}=4\mathbf{i}-3\mathbf{j}+5\mathbf{k}$ such that $|\mathbf{a}|\,=\,|\,\mathbf{b}|$ is
If a is any vector in space, then
If vectors $\mathbf{a},\,b,\,\mathbf{c}$ satisfy the condition $|\mathbf{a}-\mathbf{c}|=|\mathbf{b}-\mathbf{c}|$, then $(\mathbf{b}-\mathbf{a})\,.\,\left( \mathbf{c}-\frac{\mathbf{a}+\mathbf{b}}{\mathbf{2}} \right)$is equal to
$(\vec{a}\cdot \vec{b})\vec{c}$ and $(\vec{a}\cdot \vec{c})\vec{b}$ are
If $a=(1,\,-1,\,2),\ b=(-2,\,3,\,5)$, $\mathbf{c}=(2\,,\,-2,\,4)$ and i is the unit vector in the x-direction, then $(a-2b+3c)\,.\,i=$
For any three non-zero vectors ${{r}_{1}},\,{{r}_{2}}$ and ${{r}_{3}}$, $\begin{vmatrix} {{r}_{1}}\,.\,{{r}_{1}} & {{r}_{1}}\,.\,{{r}_{2}} & {{r}_{1}}\,.\,{{r}_{3}} \\ {{r}_{2}}\,.\,{{r}_{1}} & {{r}_{2}}\,.\,{{r}_{2}} & {{r}_{2}}\,.\,{{r}_{3}} \\ {{r}_{3}}\,.\,{{r}_{1}} & {{r}_{3}}\,.\,{{r}_{2}} & {{r}_{3}}\,.\,{{r}_{3}} \\ \end{vmatrix}=0$. Then which of the following is false
Let a, b and c be vectors with magnitudes 3, 4 and 5 respectively and a + b + c = 0, then the values of a.b + b.c + c.a is
If a and b are adjacent sides of a rhombus, then

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