VectorsEdexcel IGCSE Maths: Flashcards

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What two properties define a vector?

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What two properties define a vector?: Magnitude (size) and direction.
How do you find $\vec{AB}$ if you know the position vectors $\vec{a}$ and $\vec{b}$ of points $A$ and $B$ ?: $\vec{AB} = \vec{b} - \vec{a}$ (end minus start).
How do you add two column vectors $\begin{pmatrix} a \\ b \end{pmatrix}$ and $\begin{pmatrix} c \\ d \end{pmatrix}$ ?: Add component-wise: $\begin{pmatrix} a+c \\ b+d \end{pmatrix}$ .
What effect does multiplying a vector by scalar $k$ have on its length and direction?: Length scales by $|k|$ ; direction stays the same if $k>0$ , reverses if $k<0$ .
Formula for the magnitude of $\vec{a} = \begin{pmatrix} x \\ y \end{pmatrix}$ ?: $|\vec{a}| = \sqrt{x^2+y^2}$ (Pythagoras' theorem).
Find $|\vec{v}|$ if $\vec{v} = \begin{pmatrix} 3 \\ 4 \end{pmatrix}$ .: $\sqrt{3^2+4^2} = \sqrt{25} = 5$ .
What is a position vector?: A vector giving the location of a point relative to a fixed origin $O$ , e.g. $\vec{OA}$ .
Position vector of the midpoint $M$ of $AB$ , given $\vec{OA}=\vec{a}$ , $\vec{OB}=\vec{b}$ ?: $\vec{OM} = \tfrac{1}{2}(\vec{a}+\vec{b})$ .
Formula for the point $M$ dividing $AB$ in ratio $m:n$ from $A$ to $B$ ?: $\vec{OM} = \vec{a} + \dfrac{m}{m+n}(\vec{b}-\vec{a})$ .
When are two vectors described as parallel?: When one is a scalar multiple of the other, e.g. $\vec{b} = k\vec{a}$ .
What two conditions together prove points $A$ , $B$ , $C$ are collinear?: $\vec{AB}$ and $\vec{BC}$ must be parallel (one a scalar multiple of the other) AND share a common point (e.g. $B$ ).
If $\vec{AB} = 3\vec{a} - 2\vec{b}$ and $\vec{CD} = 6\vec{a} - 4\vec{b}$ , what is the relationship between $AB$ and $CD$ ?: They are parallel, since $\vec{CD} = 2\vec{AB}$ .
What is the relationship between $\vec{AB}$ and $\vec{BA}$ ?: $\vec{BA} = -\vec{AB}$ — same magnitude, opposite direction.
Is magnitude ever negative?: No, magnitude is always positive (or zero for the zero vector) since it is a square root of a sum of squares.
What does the zero vector $\begin{pmatrix} 0 \\ 0 \end{pmatrix}$ represent?: No displacement at all — magnitude zero and no defined direction.