If the component of the vector $\mathbf{A}$ along the vector $\mathbf{B}$ is twice the component of $\mathbf{B}$ along $\mathbf{A}$, then the ratio of magnitudes of vectors $\mathbf{A}$ and $\mathbf{B}$ is

The angle between force $\mathbf{F}=3 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}-5 \hat{\mathbf{k}}$ and displacement $\mathbf{d}=5 \hat{\mathbf{i}}+4 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}$ is

An ant starts from the origin and crawls 10 cm along the $X$-axis and then 20 cm along the $Y$-axis. The dot product of the ant's displacement vector with the position vector of a point that makes $45^{\circ}$ with the $X$-axis and has a magnitude of $\sqrt{2} \mathrm{~cm}$ is

If $\mathbf{r}_1=2 \hat{\mathbf{x}}, \mathbf{r}_2=2 \hat{\mathbf{y}}$, where $\hat{\mathbf{x}}$ and $\hat{\mathbf{y}}$ are unit vectors along the $X$-axis and $Y$-axis respectively, then the magnitude of $\mathbf{r}_1+\mathbf{r}_2$ is