Vectors

2012 NJC P1 Q2 Relative to the origin $O$, the position vectors of the points $A$, $B$ and $P$ are $-mathbf{i}-3mathbf{j}+2mathbf{k}$, $5mathbf{i}+2mathbf{k}$ and $left( 1+2lambda right)mathbf{i}+left( lambda -2 right)mathbf{j}+2mathbf{k}$ respectively,

2018 EJC Promo Q5 (a) Given that $mathbf{u}$ and $mathbf{v}$ are non-zero vectors such that $mathbf{u}times mathbf{v}=left( mathbf{u}-mathbf{v} right)times left( mathbf{u}+mathbf{v} right)$, what can be deduced about $mathbf{u}$ and $mathbf{v}$?

2022 EJC J2 MCT Q8  Planes $p$ and $q$ have equations $x+y+z=5$ and $3x+y-z=9$ respectively, and meet in the line $l$. (a) Find a vector equation for $l$. [2] (b)

2021 TMJC P1 Q9 The line $l$ passes through the point $A$ with coordinates $left( 1,-2,3 right)$ and is parallel to the vector $left( begin{matrix} 4 \ 0 \ -1

2022 MI P2 Q4 The equations of the plane ${{pi }_{1}}$ and the line $l$ are $mathbf{r}cdot left( begin{matrix} 1 \ 0 \ 1 \ end{matrix} right)=1$ and $frac{x}{2}=y+1$, $z=2$

2021 YIJC P2 Q4 The line $l$ has equation $frac{x+1}{2}=frac{y-a}{b}=frac{z-4}{3}$. The plane$~p$ has equation $x+y-z-11=0$. (i) If $l$ and $~p$ do not intersect, what can be said about the values

2023 NJC Promo Q5 (b) Two non-zero vectors $mathbf{a}$ and $mathbf{b}$ are such that $left| mathbf{a}times mathbf{b} right|=sqrt{3}$, $left| mathbf{a}-mathbf{b} right|=1$ and the angle between the vectors $mathbf{a}$ and $mathbf{b}$

2022 ACJC Promo Q10 The plane ${{P}_{1}}$ has the equation $-6x-4y+2z=4$. (i) Find the vector equations of the planes such that the perpendicular distance from each plane to ${{P}_{1}}$ is