Questions in vectors-m

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If x and y are two unit vectors and $\pi $ is the angle between them, then $\frac{1}{2}|x-y|$ is equal to
If $a\,.\,i=a\,.\,(\hat{i}+\hat{j})=a\,.\,(\hat{i}+\hat{j}+\hat{k})$, then $\vec{a} =$
If $\hat{i}, \hat{j}, \hat{k}$ are unit vectors, then
If $|a|\,=\,|\mathbf{b}|,$ then $(a+b)\,.\,(a-b)$ is
$a,\,b,\,c$ are three vectors, such that $a+b+c=0$, $|a|\,=1,\,|b|\,=2,\,|c|\,=3$, then $a.b+b.c+c.a$ is equal to
A unit vector which is coplanar to vector $\mathbf{i}+\mathbf{j}+2k$ and $\mathbf{i}+2\mathbf{j}+\mathbf{k}$ and perpendicular to $\mathbf{i}+\mathbf{j}+\mathbf{k},$ is
If $|\mathbf{a}|\,=3,\,\,|\mathbf{b}|\,=4$ then a value of ? for which $\mathbf{a}+\lambda \mathbf{b}$ is perpendicular to $\mathbf{a}-\lambda \mathbf{b}$ is
$\mathbf{a},\,\mathbf{b}$ and c are three vectors with magnitude $|\mathbf{a}|\,=4,$ $|\mathbf{b}|\,=4,$ $|\mathbf{c}|\,=2$ and such that $\mathbf{a}$ is perpendicular to $(\mathbf{b}+\mathbf{c}),\,\mathbf{b}$ is perpendicular to $(\mathbf{c}+\mathbf{a})$ and $\mathbf{c}$ is perpendicular to $(\mathbf{a}+\mathbf{b}).$ It follows that $|\mathbf{a}+\mathbf{b}+\mathbf{c}|$ is equal to
The angle between the vectors $3\,\mathbf{i}+\mathbf{j}+2\,\mathbf{k}$ and $2\,\mathbf{i}-2\,\mathbf{j}+4\,\mathbf{k}$ is
If the position vectors of the points A, B, C, D be $\mathbf{i}+\mathbf{j}+\mathbf{k},\,\,2\,\mathbf{i}+5\,\mathbf{j},\,\,3\,\mathbf{i}+2\,\mathbf{j}-3\mathbf{k}$and $\mathbf{i}-6\,\mathbf{j}-\mathbf{k},$ then the angle between the vectors $\overrightarrow{AB}$ and $\overrightarrow{CD}$ is

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