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 $|\vec{u}|$ equals:

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

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.