Vectors (Edexcel IGCSE Further Maths): Exam Questions

$O,A$ and $B$ are fixed points such that

$\overset{\rightarrow}{OA}=4i+3j\overset{\rightarrow}{OB}=8i+pj$ and $\left| \vec{AB} \right| = 2 \sqrt{13}$

Find the possible values of $p$.

Given that $p>0$

find a unit vector parallel to $\overset{\rightarrow}{AB}$

The points $A,B,C$ and $D$ are the vertices of a quadrilateral such that

$\overset{\rightarrow}{AB}=3a+4b\overset{\rightarrow}{AC}=7a+9b\overset{\rightarrow}{AD}=4a+5b$

Show that $ABCD$is a parallelogram.

$BC$ is extended to the point $E$ such that $BCE$ is a straight line.

Point $F$ lies on $CD$ such that $CF:FD=1:2$

Given that $A,F$ and $E$ are collinear,

find the vector $\overset{\rightarrow}{AE}$ in the form $Xa+Yb$ where $X$and $Y$are rational numbers to be found.

Figure 2 shows triangle $AOB$

$\overset{\rightarrow}{OA}=4a+5b\overset{\rightarrow}{OB}=8a-b\overset{\rightarrow}{OD}=15a+10b$ where $\vert a \vert = \vert b \vert = 1$

(i) Find $\overset{\rightarrow}{AB}$ in terms of $a$ and $b$

(ii) Find, in its simplest form, the exact value of $\vert \vec{AB} \vert$

Figure 2 shows triangle $AOB$

$\overset{\rightarrow}{OA}=4a+5b\overset{\rightarrow}{OB}=8a-b\overset{\rightarrow}{OD}=15a+10b$ where $\vert a \vert = \vert b \vert = 1$

Find the area of triangle $AOB$

Figure 2 shows triangle $AOB$

$\overset{\rightarrow}{OA}=4a+5b\overset{\rightarrow}{OB}=8a-b\overset{\rightarrow}{OD}=15a+10b$ where $\vert a \vert = \vert b \vert = 1$

The point $C$ lies on $AB$ and $OD$ such that $O$, $C$ and $D$ are collinear.

Use a vector method to find vector $\overset{\rightarrow}{OC}$ as a simplified expression in terms of $a$ and $b$

$O$, $A$ and $B$ are fixed points such that

$\overset{\rightarrow}{OA}=(b+1)i+bj$ $\overset{\rightarrow}{AB}=3i$ The unit vector parallel to $\overset{\rightarrow}{OB}$ is $\frac{\sqrt{17}}{34} [ ( 3 a + 2 ) i + b j ]$

Given that $a$ and $b$ re constants where $a>0$ and $b>0$

find the exact value of

(i) $a$

(ii) $b$

Figure 3 shows quadrilateral $OABC$ where

$\overset{\rightarrow}{OA}=4p+5q\overset{\rightarrow}{OB}=3p+q\overset{\rightarrow}{OC}=2p-4q$

The point $M$ is the midpoint of $OC$

Find $\overset{\rightarrow}{MA}$ as a simplified expression in terms of $p$ and $q$

The point $N$ lies on $OB$ such that $M$, $N$ and $A$are collinear.

Find the ratio $MN:NA$

Figure 3 shows triangle $OAB$ with $\overset{\rightarrow}{OA}=a$ and $\overset{\rightarrow}{OB}=b$

$M$ is the midpoint of $OA$.

$N$ is the point on $OB$ such that $ON:NB=3:1$

The lines $AN$ and $BM$ intersect at the point $X$.

Find expressions, in terms of $a$and $b$, for

(i) $\overset{\rightarrow}{AN}$

(ii) $\overset{\rightarrow}{BM}$

Using a vector method, find $AX:XN$