P4: VectorsEdexcel International A Level Maths: Topic test

1. 1
   The vectors $\mathbf{u}=3\mathbf{i}-4\mathbf{j}+12\mathbf{k}$ and $\mathbf{w}=2\mathbf{i}+\mathbf{j}-2\mathbf{k}$.

   (a)

   Find $|\mathbf{u}|$.

   [1 mark]

   • A$5$
   • B$13$
   • C$19$
   • D$169$

   (b)

   Which of the following is the unit vector in the direction of $\mathbf{w}$?

   [1 mark]

   • A$\frac19\left(2\mathbf{i}+\mathbf{j}-2\mathbf{k}\right)$
   • B$\frac15\left(2\mathbf{i}+\mathbf{j}-2\mathbf{k}\right)$
   • C$\frac13\left(2\mathbf{i}+\mathbf{j}+2\mathbf{k}\right)$
   • D$\frac13\left(2\mathbf{i}+\mathbf{j}-2\mathbf{k}\right)$

   (c)

   Find a vector that is parallel to $\mathbf{w}$ and has magnitude 15.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The line $l$ has equation $\mathbf{r}=(\mathbf{i}-2\mathbf{j}+4\mathbf{k})+\lambda(2\mathbf{i}+\mathbf{j}-3\mathbf{k})$.

   (a)

   Which of the following points lies on $l$?

   [1 mark]

   • A$(3,-1,7)$
   • B$(2,-1,1)$
   • C$(3,-1,1)$
   • D$(5,0,-1)$

   (b)

   Which of the following lines is parallel to $l$?

   [1 mark]

   • A$\mathbf{r}=(5\mathbf{j}+\mathbf{k})+\mu(4\mathbf{i}+2\mathbf{j}-6\mathbf{k})$
   • B$\mathbf{r}=(\mathbf{i}-2\mathbf{j}+4\mathbf{k})+\mu(2\mathbf{i}+\mathbf{j}+3\mathbf{k})$
   • C$\mathbf{r}=(2\mathbf{i}+\mathbf{j}-3\mathbf{k})+\mu(\mathbf{i}-2\mathbf{j}+4\mathbf{k})$
   • D$\mathbf{r}=(\mathbf{i}-2\mathbf{j}+4\mathbf{k})+\mu(4\mathbf{i}+2\mathbf{j}-3\mathbf{k})$

   (c)

   The point $P(9,2,p)$ lies on $l$. Find the value of $p$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The line $l_1$ has equation $\mathbf{r}=(\mathbf{i}+2\mathbf{j}+\mathbf{k})+s(2\mathbf{i}-\mathbf{j}+\mathbf{k})$ and the line $l_2$ has equation $\mathbf{r}=(2\mathbf{i}+4\mathbf{k})+t(\mathbf{i}+\mathbf{j}-2\mathbf{k})$.

   (a)

   Show that $l_1$ and $l_2$ intersect, and find the position vector of the point of intersection $P$.

   [3 marks]

   (b)

   The points $A$ and $B$ lie on $l_1$ and $l_2$ respectively, where $A$ corresponds to $s=0$ and $B$ corresponds to $t=0$. Find the size of angle $APB$, giving your answer to 1 decimal place.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The line $l$ has equation $\mathbf{r}=(4\mathbf{i}+\mathbf{j}+2\mathbf{k})+\lambda(\mathbf{i}-2\mathbf{j}+2\mathbf{k})$. The point $Q$ has coordinates $(7,1,5)$, and the point $F$ lies on $l$ such that $QF$ is perpendicular to $l$.

   (a)

   Find the coordinates of $F$, and find the distance $QF$.

   [6 marks]

   (b)

   The point $R$ lies on $l$ and $QR=3\sqrt2$. Find the two possible position vectors of $R$.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   Relative to the origin $O$, the points $P$ and $Q$ have coordinates $(2,-3,6)$ and $(-1,1,6)$.

   (a)

   Find the distance $PQ$.

   [1 mark]

   • A$5$
   • B$25$
   • C$7$
   • D$\sqrt5$

   (b)

   Find $|\overrightarrow{OP}|$.

   [1 mark]

   • A$49$
   • B$11$
   • C$7$
   • D$\sqrt{13}$

   (c)

   The point $R$ is such that $OPRQ$ is a parallelogram. Find the exact value of $|\overrightarrow{OR}|$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   The vectors $\mathbf{a}=4\mathbf{i}+p\mathbf{j}-2\mathbf{k}$ and $\mathbf{b}=\mathbf{i}-2\mathbf{j}+3\mathbf{k}$, where $p$ is a constant.

   (a)

   Given that $\mathbf{a}$ and $\mathbf{b}$ are perpendicular, find $p$.

   [1 mark]

   • A$1$
   • B$5$
   • C$-5$
   • D$-1$

   (b)

   When $p=2$, find the exact value of $\cos\theta$, where $\theta$ is the angle between $\mathbf{a}$ and $\mathbf{b}$.

   [1 mark]

   • A$\frac{3}{2\sqrt{21}}$
   • B$-\frac{3}{2\sqrt{21}}$
   • C$-\frac{6}{\sqrt{24}+\sqrt{14}}$
   • D$-\frac{1}{56}$

   (c)

   Hence find the angle between $\mathbf{a}$ and $\mathbf{b}$ when $p=2$, giving your answer to 1 decimal place.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   The line $l_1$ has equation $\mathbf{r}=(\mathbf{i}+2\mathbf{k})+s(\mathbf{i}+2\mathbf{j}-\mathbf{k})$ and the line $l_2$ has equation $\mathbf{r}=(3\mathbf{i}+\mathbf{j})+t(2\mathbf{i}-\mathbf{j}+4\mathbf{k})$.

   (a)

   Show that $l_1$ and $l_2$ are skew lines.

   [3 marks]

   (b)

   Find the acute angle between $l_1$ and $l_2$, giving your answer to 1 decimal place.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   The line $l_1$ passes through the points $A(3,-2,1)$ and $B(7,0,5)$. The line $l_2$ has equation $\mathbf{r}=(4\mathbf{i}+\mathbf{j}+\mathbf{k})+\mu(\mathbf{i}-2\mathbf{j}+2\mathbf{k})$.

   (a)

   (i) Find a vector equation of $l_1$.
   (ii) Show that $l_1$ and $l_2$ intersect, and find the coordinates of the point of intersection $C$.

   [6 marks]

   (b)

   (i) Find the acute angle between $l_1$ and $l_2$, giving your answer to 1 decimal place.
   (ii) The point $D$ lies on $l_2$ with $\mu=3$. Find the shortest distance from $D$ to $l_1$, giving your answer to 3 significant figures.

   [6 marks]

   Total for question 8: 12 marks

End of questions