Vectors in the Plane practice questions — A-Level Maths

Are the vectors $\vec{u}=\vec{i}+2\vec{j}$ and $\vec{v}=2\vec{i}+4\vec{j}$ collinear?

For $\vec{u}=\vec{i}+\vec{j}$ and $\vec{v}=a\vec{i}-2\vec{j}$, find $a\in \mathbb{R}$ so that $\vec{u}$ and $\vec{v}$ are collinear.

The vector $\overrightarrow{AB}$ from $A(1,2)$ to $B(4,6)$ equals:

Find $m\in \mathbb{R}$ so that $\vec{u}=m\vec{i}+3\vec{j}$ and $\vec{v}=4\vec{i}+(m+1)\vec{j}$ are perpendicular.

For $\vec{u}=\vec{i}+2\vec{j}$ and $\vec{v}=3\vec{i}-\vec{j}$, the sum $\vec{u}+\vec{v}$ equals:

The dot product $\vec{u}\cdot \vec{v}$ for $\vec{u}=2\vec{i}+3\vec{j}$, $\vec{v}=\vec{i}-4\vec{j}$ equals:

For $\vec{u}=3\vec{i}-2\vec{j}$, the vector $2\vec{u}$ equals:

For $\vec{u}=3\vec{i}+4\vec{j}$, the magnitude $\mathrm{\mid }\vec{u}\mathrm{\mid }$ equals:

For $\vec{u}=\vec{i}+\vec{j}$ and $\vec{v}=a\vec{i}-2\vec{j}$, find $a\in \mathbb{R}$ so that $\mathrm{\mid }\vec{u}+\vec{v}{\mathrm{\mid }}^2=\mathrm{\mid }\vec{u}{\mathrm{\mid }}^2+\mathrm{\mid }\vec{v}{\mathrm{\mid }}^2$.

Let $\vec{a}$ and $\vec{b}$ be two non-collinear vectors. Find $m\in \mathbb{R}$ so that $\vec{u}=3\vec{a}-(m+1)\vec{b}$ and $\vec{v}=(m-1)\vec{a}-5\vec{b}$ are collinear.