Let $$\mathbf{a}=x \hat{i}+y \hat{j}+z \hat{k}$$ and $$x=2 y$$. If $$|\mathbf{a}|=5 \sqrt{2}$$ and a makes an angle of $$135^{\circ}$$ with the Z-axis, then $$\mathbf{a}=$$

Let $$\mathbf{a}, \mathbf{b}, \mathbf{c}$$ be the position vectors of the vertices of a $$\triangle A B C$$. Through the vertices, lines are drawn parallel to the sides to form the $$\Delta A^{\prime} B^{\prime} C^{\prime}$$. Then, the centroid of $$\Delta A^{\prime} B^{\prime} C^{\prime}$$ is

The position vectors of the points $$A$$ and $$B$$ with respect to $$O$$ are $$2 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+\hat{\mathbf{k}}$$ and $$2 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}+4 \hat{\mathbf{k}}$$. The length of the internal bisector of $$\angle B O A$$ of $$\triangle A O B$$ is (take proportionality constant is 2)

Let $$\mathbf{u}=2 \hat{\mathbf{i}}+3 \hat{\mathbf{j}}+\hat{\mathbf{k}}, \mathbf{v}=-3 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}$$ and $$\mathbf{w}=\hat{\mathbf{i}}-\hat{\mathbf{j}}+4 \hat{\mathbf{k}}$$. Then which of the following statement is true?