Applications of vectors to 3-D geometryEdexcel A-Level Further Maths: Subtopic test

1. 1
   The line $l$ has vector equation $\mathbf{r}=(\mathbf{i}-2\mathbf{j}+3\mathbf{k})+\lambda(2\mathbf{i}+\mathbf{j}-2\mathbf{k})$.

   (a)

   Find the direction cosines of $l$, taking the direction given by the vector equation.

   [1 mark]

   • A$\frac23,\ \frac13,\ -\frac23$
   • B$2,\ 1,\ -2$
   • C$\frac25,\ \frac15,\ -\frac25$
   • D$\frac49,\ \frac19,\ \frac49$

   (b)

   Which of these is an equation of $l$ in the form $(\mathbf{r}-\mathbf{a})\times\mathbf{b}=\mathbf{0}$?

   [1 mark]

   • A$\left(\mathbf{r}-(2\mathbf{i}+\mathbf{j}-2\mathbf{k})\right)\times(\mathbf{i}-2\mathbf{j}+3\mathbf{k})=\mathbf{0}$
   • B$\left(\mathbf{r}+(\mathbf{i}-2\mathbf{j}+3\mathbf{k})\right)\times(2\mathbf{i}+\mathbf{j}-2\mathbf{k})=\mathbf{0}$
   • C$\left(\mathbf{r}-(\mathbf{i}-2\mathbf{j}+3\mathbf{k})\right)\times(2\mathbf{i}+\mathbf{j}-2\mathbf{k})=\mathbf{0}$
   • D$\left(\mathbf{r}-(\mathbf{i}-2\mathbf{j}+3\mathbf{k})\right)\cdot(2\mathbf{i}+\mathbf{j}-2\mathbf{k})=0$

   (c)

   Find Cartesian equations for $l$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The plane $\Pi$ passes through the point $A(1,2,3)$ and is perpendicular to the vector $2\mathbf{i}-\mathbf{j}+2\mathbf{k}$.

   (a)

   Find a Cartesian equation of $\Pi$.

   [1 mark]

   • A$2x-y+2z=0$
   • B$x+2y+3z=6$
   • C$2x-y+2z=2$
   • D$2x-y+2z=6$

   (b)

   Find the perpendicular distance from the point $(4,3,-1)$ to $\Pi$.

   [1 mark]

   • A$-1$
   • B$1$
   • C$3$
   • D$\frac13$

   (c)

   The line $m$ passes through the origin and has direction $\mathbf{i}+\mathbf{j}+\mathbf{k}$. Find the coordinates of the point where $m$ meets $\Pi$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The line $l$ has Cartesian equation $\frac{x-1}{3}=\frac{y+2}{-4}=\frac{z}{12}$, and the point $P$ has coordinates $(5,1,0)$.

   (a)

   Find the direction cosines of $l$, and the acute angle that $l$ makes with the $z$-axis.

   [3 marks]

   (b)

   Find the shortest distance from $P$ to $l$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The line $l$ has vector equation $\mathbf{r}=(\mathbf{i}+\mathbf{j})+\lambda(2\mathbf{i}+\mathbf{j}-\mathbf{k})$ and the plane $\Pi$ has equation $\mathbf{r}\cdot(\mathbf{i}+2\mathbf{j}+2\mathbf{k})=9$.

   (a)

   (i) Find the coordinates of the point where $l$ meets $\Pi$.
   (ii) Find the acute angle between $l$ and $\Pi$.

   [6 marks]

   (b)

   The point $A$ has position vector $\mathbf{i}+\mathbf{j}$ and lies on $l$. Find the coordinates of the foot of the perpendicular from $A$ to $\Pi$, and hence the perpendicular distance from $A$ to $\Pi$.

   [6 marks]

   Total for question 4: 12 marks

End of questions