If the volume of the parallelopiped formed by the vectors $$\hat{\mathbf{i}}+a \hat{\mathbf{j}}+\hat{\mathbf{k}}, \hat{\mathbf{j}}+a \hat{\mathbf{k}}$$ and $$a \hat{\mathbf{i}}+\hat{\mathbf{k}}$$ becomes minimum, then $$a$$ is equal to

If $$\mathbf{a}=\frac{3}{2} \hat{\mathbf{k}}$$ and $$\mathbf{b}=\frac{2 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}-\hat{\mathbf{k}}}{2}$$, then angle between $$\mathbf{a}+\mathbf{b}$$ and $$\mathbf{a}-\mathbf{b}$$ is

Let $$\mathbf{a}=\hat{\mathbf{i}}+\hat{\mathbf{j}}+\hat{\mathbf{k}}, \mathbf{b}=\hat{\mathbf{i}}+3 \hat{\mathbf{j}}+5 \hat{\mathbf{k}}$$ and $$\mathbf{c}=7 \hat{\mathbf{i}}+9 \hat{\mathbf{j}}+11 \hat{\mathbf{k}}$$, then the area of parallelogram having diagonals $$\mathbf{a}+\mathbf{b}$$ and $$\mathbf{b}+\mathbf{c}$$ is

If $$\mathbf{a}$$ and $$\mathbf{b}$$ are two vectors such that $$|\mathbf{a}|=2, |\mathbf{b}|=3$$ and $$\mathbf{a}+t \mathbf{b}$$ and $$\mathbf{a}-t \mathbf{b}$$ are perpendicular, where $$t$$ is a positive scalar, then