Vectors

$\left(\genfrac{}{}{0px}{}{3}{-2}\right)$ means 3 right and 2 down. Add vectors by adding components; multiply by a number by scaling both components:

The magnitude (length) of a vector is Pythagoras on its components:

Magnitude answers are lengths — leave them as simplified surds or round as instructed, and never drop the square root.

Geometry questions define a shape with two base vectors — say $\overrightarrow{OA}=\mathbf{a}$ and $\overrightarrow{OB}=\mathbf{b}$ — and ask for other vectors in terms of $\mathbf{a}$ and $\mathbf{b}$. The method never changes: walk from the start letter to the end letter along edges you know, writing each leg with its sign. Walking against an arrow negates it:

Ratios scale the legs. If $M$ lies on $OA$ with $OM:MA=2:1$, then $\overrightarrow{OM}=\frac{2}{3}\mathbf{a}$ — the fraction is part over whole, 2 parts of 3.

The position vector of a point is its vector from the origin $O$. The midpoint $M$ of $AB$ has position vector

— the average of the endpoints’ position vectors, exactly like a coordinate midpoint.

Two vectors are parallel when one is a scalar multiple of the other: $\mathbf{u}=k\mathbf{v}$. To prove three points $A,B,C$ are collinear (on one straight line): show $\overrightarrow{AB}=k\overrightarrow{BC}$ for some number $k$, then say the two vectors are parallel and share the point $B$. Both halves of that sentence are needed for the mark — parallel alone doesn’t put the points on one line.

1. 1.Dropping the minus when walking against an arrow

   $\overrightarrow{AO}=-\mathbf{a}$, not $\mathbf{a}$. Before combining anything, write the route letter-by-letter and attach signs leg by leg — don’t do it in your head.

2. 2.Order-of-operations slips in expressions like a − 2b

   Scale first, then add: $\mathbf{a}-2\mathbf{b}$ means $\mathbf{a}+(-2\mathbf{b})$. Examiners specifically note candidates mangling exactly this shape.

3. 3.Ratio fractions using part-over-part

   $OM:MA=2:1$ puts $M$ at $\frac{2}{3}$ of the way along — 2 parts out of 3 total, never $\frac{2}{1}$ or $\frac{1}{2}$.

4. 4.Claiming collinearity from parallelism alone

   Finish the sentence: the vectors are parallel and pass through a common point. Without the shared point the proof mark is withheld.

5. 5.Magnitude without the square root

   $\mathrm{\mid }\left(\genfrac{}{}{0px}{}{6}{-8}\right)\mathrm{\mid }=\sqrt{36+64}=10$ — candidates regularly stop at 100. If your “length” looks suspiciously huge, you forgot the root.