What is the area of the parallelogram whose sides are represented by the vectors $\hat{i}+2\hat{j}+3\hat{k}$ and $2\hat{i}+\hat{j}+2\hat{k}$ ?

The position vectors of the vertices A, B, Cand D of a quadrilateral ABCD are given by $3\hat{i}+4\hat{j}-2\hat{k}$ , $4\hat{i}-4\hat{j}-3\hat{k}$, $2\hat{i}-3\hat{j}+2\hat{k}$ and $6\hat{i}-2\hat{j}+\hat{k}$ respectively.
What is the angle between the diagonals AC and BD of the quadrilateral?

A force$\vec{F} = 2\hat{i} - \lambda\hat{j} + 5\hat{k}$ is applied at the point A(1, 2, 5) If its moment about the point B(- 1, - 2, 3) is $16\hat{i} - 6\hat{j} + 2\lambda\hat{k}$, then what is the value of $\lambda$ ?

A line makes angles α, β and γ with the positive directions of the coordinate axes. If $\$\backslash vec\{a\}=(\backslash sin^2\; \backslash alpha)\backslash hat\{i\}\; +\; (\backslash sin^2\; \backslash beta)\backslash hat\{j\}\; +\; (\backslash sin^2\; \backslash gamma)\backslash hat\{k\}\; \backslash text\{\; and\; \}\; \backslash vec\{b\}\; =\; \backslash hat\{i\}\; +\; \backslash hat\{j\}\; +\; \backslash hat\{k\}\$$ , then what is $\vec{a}.\vec{b}$ equal to?