Vectors

The position vectors of A, B and C relative to origin O are $-\mathbf i+5\mathbf j+\mathbf k$, $\mathbf i-\mathbf j+5\mathbf k$ and $p\mathbf i-4\mathbf j+7\mathbf k$ respectively, where $p$ is constant. Given A, B and C are collinear, show $p=2$. ****

Relative to origin O, P and Q have position vectors $\mathbf p,\mathbf q$, nonparallel and nonzero. R lies on PQ produced with $PQ:QR=1:\lambda$. M is the midpoint of OR.

Lines $l_1,l_2$ are $\mathbf r=(1,0,1)+\lambda(1,1,-2)$ and $\mathbf r=(1,2,-1)+\mu(7,5,1)$ respectively.

Relative to origin O, A and B have position vectors $\mathbf a=3\mathbf i+\mathbf j+3\mathbf k$ and $\mathbf b=5\mathbf i-4\mathbf j+3\mathbf k$.

A and B have position vectors $\mathbf a,\mathbf b$ relative to O, with $|\mathbf a|=|\mathbf b|$. P with position vector $\mathbf p$ lies on AB such that $\mathbf b\cdot\mathbf p=\mathbf a\cdot\mathbf p$.

Relative to O, A and B have position vectors a,b, nonzero and nonparallel.

(i) State the shortest distance from B to the line through O,A. ****

Line l passes through A$(1,2,-1)$ and B$(2,a,1)$. Given l cuts the x-axis, show $a=-2$. **** Find the shortest distance of O from l. ****

A,B,C have position vectors a,b,c relative to O. $\overrightarrow{AC}=4\overrightarrow{CB}$ and $|\mathbf a+\mathbf b|^2=|\mathbf a|^2+|\mathbf b|^2$.

Relative to O, A,B,D have position vectors a,b,d. OACB is a parallelogram and $\angle OAC=2\pi/3$ radians.

Planes $\Pi_1,\Pi_2$ are $\mathbf r\cdot(2,4,1)=10$ and $\mathbf r\cdot(1,3,1)=8$.

Line l passes through A$(-1,2,3)$ and B$(5,14,11)$. Plane p is $2x+3y-6z=-7$.

Plane $\pi_1$ and line l are $\mathbf r\cdot(2,1,-2)=6$ and $(x-1)/3=(y+4)/2=z-5$ respectively. A has position vector $3\mathbf i-\mathbf j+4\mathbf k$.

A plane $\pi_1$ has equation $\mathbf r\cdot(2,-1,3)=9$ and a line $l_1$ has equation $\mathbf r=(3,0,0)+\lambda(4,1,-2)$.

The diagram above shows a cuboid with horizontal rectangular base $ABCD$, where $AB=5$ units, $AD=3$ units and $AP=2$ units. The edges $AP,BQ,CR,DS$ are vertical, and $PQRS$ is the top of the cuboid. The point $A$ is taken as the origin and vectors $\mathbf i,\mathbf j,\mathbf k$, each of length 1 unit, are taken along $AB,AD,AP$ respectively.

The plane $p_1$ has equation $x+y-2z=4$. A plane $p_2$ with equation $2x+ay+bz=-4$ (where $a,b$ are constants) is parallel to $p_1$, find the values of $a,b$ and also the exact distance between $p_1,p_2$.

Relative to the origin $O$, the point $A$ has position vector $\overrightarrow{OA}=(0,1,0)$ and $A$ lies on the plane $\Pi_1$ with equation defined by $\mathbf r\cdot(1,3,2)=3$. Another plane $\Pi_2$ has equation $y=x$. The planes $\Pi_1,\Pi_2$ intersect at line $l$.

The equations of the line $l_1$ and the plane $p_1$ are respectively, $\mathbf r=(1,0,0)+\mu(-1,1,1)$ and $\mathbf r=s(1,1,1)+t(2,-1,1)$, where $\mu,s,t\in\mathbb R$.

The points $A,B$ have position vectors $3\mathbf i+\mathbf j$ and $3\mathbf i+3\mathbf j$ respectively. The line $l_1$ and the planes $\Pi_1,\Pi_2$ have equations as follows:

A home interior designer is looking to design a pull-down wardrobe lift. To test his concept, the designer built a small-scale model of it, in which points $(x,y,z)$ are defined relative to a corner of the model at $O(0,0,0)$. The rods used in the model can be modelled by equations of straight lines.

Laser (Light Amplification by Stimulated Emission of Radiation) has many applications including medicine, data storage, military and industrial uses. It has the property of spatial coherence, which allows the laser beam to stay narrow over long distances. When a laser beam is projected onto a mirror at an angle, it reflects off the mirror at the same angle.

Peter is using equations of planes to model two hillsides that meet along a river. The river is modelled by the line where the two planes meet.

Figure 1 shows a methane molecule consisting of a carbon atom with four hydrogen atoms symmetrically placed around it. Figure 2 shows the tetrahedron structure of the methane molecule with the centres of the hydrogen atoms represented by points $Q,R,S,T$ and the centre of the carbon atom represented by point $V$.

The diagram above shows a partial design of the roof of the gallery of an amphitheatre. $ABCD$ is an inclined rectangular roof, where $AB=10$ m and $BC=5$ m. $EFGH$ is a rectangle on a horizontal ground, where $EF=10$ m and $EH=4$ m. The points $A,B$ are 3 m directly above $E,F$ respectively. The points $C,D$ are 6 m directly above $G,H$ respectively.