Vectors
Extension Problems
$\overrightarrow{OP}=\vec{a}, \overrightarrow{OA}=3\vec{a}$ and $\overrightarrow{OB}=\vec{b}$. $M$ is the midpoint of $AB$.
$X$ is to the left of $AO$ such that $MX$ is parallel to $BO$. The intersection of $BX$ and $AO$ is $P$.
Find $k$ for $\overrightarrow{BX}=k\overrightarrow{BP}$.
$k=$
$X$ is to the left of $AO$ such that $MX$ is parallel to $BO$. The intersection of $BX$ and $AO$ is $P$.
Find $k$ for $\overrightarrow{BX}=k\overrightarrow{BP}$.
$k=$
Consider quadrilateral $ABCD$ and point $P$ on the same plane. Let $\vec{AB}=\vec{a}, \vec{BC}=\vec{b}$ and $\vec{CD}=\vec{c}$.
If $\vec{PA}+\vec{PB}+\vec{PC}+\vec{PD}=\vec{AD}$, find $\vec{AP}$ in terms of $\vec{a}, \vec{b}$ and $\vec{c}$.
$\vec{AP}=$
If $\vec{PA}+\vec{PB}+\vec{PC}+\vec{PD}=\vec{AD}$, find $\vec{AP}$ in terms of $\vec{a}, \vec{b}$ and $\vec{c}$.
$\vec{AP}=$
Consider parallelogram $ABCD$. Let the midpoint of $BC$ and $CD$ be $E$ and $F$ respectively.
If $\vec{AE}=\vec{u}$ and $\vec{AF}=\vec{v}$, find $\vec{AB}$ and $\vec{AD}$ in terms of $\vec{u}$ and $\vec{v}$.
$\vec{AB}=$
$\vec{AD}=$
If $\vec{AE}=\vec{u}$ and $\vec{AF}=\vec{v}$, find $\vec{AB}$ and $\vec{AD}$ in terms of $\vec{u}$ and $\vec{v}$.
$\vec{AB}=$
$\vec{AD}=$
Let $\vec{a}=\binom{-3}{2}$ and $\vec{b}=\binom{2}{1}$.
Find the value of $t$ that gives the minimum value of $\left \vert \vec{a}+t\vec{b} \right \vert$.
$t=$
minimum value of $\left \vert \vec{a}+t\vec{b} \right \vert =$
Find the value of $t$ that gives the minimum value of $\left \vert \vec{a}+t\vec{b} \right \vert$.
$t=$
minimum value of $\left \vert \vec{a}+t\vec{b} \right \vert =$
$\vec a=\binom{2}{4}$ and $\vec b=\binom{x}{1}$. Find $x$ if $\vec a + 2\vec b$ is parallel to $2\vec a – \vec b$.
$x=$
Consider quadrilateral $ABCD$. Let $P$ be the midpoint of diagonal $AC$ and let $Q$ be the midpoint of diagonal $BD$. If $\overrightarrow{BC}=\vec a$ and $\overrightarrow{DA}=\vec b$, find $\overrightarrow{PQ}$ in terms of $\vec a$ and $\vec b$.
For $\triangle OAB$, point $C$ lies on $OA$ such that $OC:CA=1:2$ and point $D$ lies on $OB$ such that $OD:DB=3:2$.
Point $E$ is taken so that $\overrightarrow{AE}=\frac{5}{3} \overrightarrow{AD}$. Point $F$ is the intersection of line $OE$ and $BC$. Let $\vec a = \overrightarrow{OA}$ and $\vec b = \overrightarrow{OB}$.
a) Find $\overrightarrow{OE}$ in terms of $\vec a$ and $\vec b$.
b) Find $\overrightarrow{OF}$ in terms of $\vec a$ and $\vec b$.
c) Find the value of $\displaystyle \frac{FC}{CB}$.
Point $E$ is taken so that $\overrightarrow{AE}=\frac{5}{3} \overrightarrow{AD}$. Point $F$ is the intersection of line $OE$ and $BC$. Let $\vec a = \overrightarrow{OA}$ and $\vec b = \overrightarrow{OB}$.
a) Find $\overrightarrow{OE}$ in terms of $\vec a$ and $\vec b$.
b) Find $\overrightarrow{OF}$ in terms of $\vec a$ and $\vec b$.
c) Find the value of $\displaystyle \frac{FC}{CB}$.
$\overrightarrow{OE}=$
$\overrightarrow{OF}=$
$\phantom{}\frac{FC}{CB}=$
If $\vec a \ne \vec 0, \vec b \ne \vec 0$ and $\left|\vec a\right|=\left|\vec b\right|=\left|\vec a + \vec b\right|$,
a) Find the angle between $\vec a$ and $\vec b$ in degrees.
*You don’t need to include the $^\circ$ symbol
b) Find the exact value of $k$ for $\left|\vec a – \vec b\right|=k\left|\vec a\right|$.
a) Find the angle between $\vec a$ and $\vec b$ in degrees.
*You don’t need to include the $^\circ$ symbol
b) Find the exact value of $k$ for $\left|\vec a – \vec b\right|=k\left|\vec a\right|$.
a)
b)
The points $\left(3,7\right)$, $\left(6,2\right)$ and $\left(2,k\right)$ are the vertices of a triangle.
List all the values of $k$ such that the triangle is a right triangle.
*List the values of $k$ in ascending order, separated by commas.