Vectors: past paper questions

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• Question 5May/June 2026, paper 11Structured11 marks

  The lines $l_1$ and $l_2$ have equations $\mathbf{r}=-2\mathbf{i}+4\mathbf{j}+4\mathbf{k}+\lambda (3\mathbf{i}-4\mathbf{j}+2\mathbf{k})$ and $\mathbf{r}=-3\mathbf{i}-2\mathbf{j}+4\mathbf{k}+\mu (2\mathbf{i}+4\mathbf{j}+3\mathbf{k})$ respectively.

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• Question 5Oct/Nov 2025, paper 11Structured11 marks

  The plane ${\mathrm{\Pi }}_1$ has equation $\mathbf{r}=-3\mathbf{i}-\mathbf{j}-\mathbf{k}+\lambda (\mathbf{j}+2\mathbf{k})+\mu (\mathbf{i}+3\mathbf{j}+\mathbf{k})$. Find an equation for ${\mathrm{\Pi }}_1$ in the form $ax+by+cz=d$.

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• Question 7Oct/Nov 2024, paper 11Structured15 marks

  The lines $l_1$ and $l_2$ have equations $\mathbf{r}=\mathbf{i}+3\mathbf{j}-2\mathbf{k}+\lambda (2\mathbf{i}+\mathbf{j}+\mathbf{k})$ and $\mathbf{r}=\mathbf{i}-2\mathbf{j}+9\mathbf{k}+\mu (\mathbf{i}-4\mathbf{j}+2\mathbf{k})$ respectively.

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• Question 6Oct/Nov 2025, paper 12Structured14 marks

  The plane $\mathrm{\Pi }$ has equation $x+3y+2z=1$. Find the perpendicular distance from the origin $O$ to the plane $\mathrm{\Pi }$.

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• Question 2Oct/Nov 2024, paper 12Structured7 marks

  The line $l_1$ has equation $\mathbf{r}=\mathbf{i}+3\mathbf{j}-\mathbf{k}+\lambda (\mathbf{i}-\mathbf{j}-4\mathbf{k})$. The plane $\mathrm{\Pi }$ contains $l_1$ and is parallel to the vector $2\mathbf{i}+5\mathbf{j}-4\mathbf{k}$.

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• Question 5Oct/Nov 2023, paper 12Structured13 marks

  The plane ${\mathrm{\Pi }}_1$ has equation $\mathbf{r}=\mathbf{i}-\mathbf{j}-2\mathbf{k}+\lambda (\mathbf{i}-2\mathbf{j}-3\mathbf{k})+\mu (3\mathbf{i}-\mathbf{k})$. Find an equation for ${\mathrm{\Pi }}_1$ in the form $ax+by+cz=d$.

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• Question 7May/June 2026, paper 13Structured16 marks

  The plane $\mathrm{\Pi }$ has equation $\mathbf{r}=\mathbf{j}+2\mathbf{k}+s(\mathbf{i}-\mathbf{j})+t(-3\mathbf{i}+\mathbf{j}-\mathbf{k})$. Obtain an equation of $\mathrm{\Pi }$ in the form $ax+by+cz=d$.

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• Question 7Oct/Nov 2024, paper 13Structured15 marks

  The lines $l_1$ and $l_2$ have equations $\mathbf{r}=\mathbf{i}+3\mathbf{j}-2\mathbf{k}+\lambda (2\mathbf{i}+\mathbf{j}+\mathbf{k})$ and $\mathbf{r}=\mathbf{i}-2\mathbf{j}+9\mathbf{k}+\mu (\mathbf{i}-4\mathbf{j}+2\mathbf{k})$ respectively.

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• Question 7May/June 2023, paper 11Structured14 marks

  The plane ${\mathrm{\Pi }}_1$ has equation $\mathbf{r}=-4\mathbf{j}-3\mathbf{k}+\lambda (\mathbf{i}-\mathbf{j}+\mathbf{k})+\mu (\mathbf{i}+\mathbf{j}-\mathbf{k})$. Obtain an equation of ${\mathrm{\Pi }}_1$ in the form $px+qy+rz=d$.

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• Question 5May/June 2025, paper 13Structured11 marks

  The plane $\mathrm{\Pi }$ has equation $\mathbf{r}=2\mathbf{i}+3\mathbf{j}-2\mathbf{k}+\lambda (\mathbf{i}-2\mathbf{j}-\mathbf{k})+\mu (3\mathbf{i}+2\mathbf{j}-2\mathbf{k})$. Find a Cartesian equation of $\mathrm{\Pi }$, giving your answer in the form $ax+by+cz=d$.

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• Question 5May/June 2024, paper 13Structured10 marks

  The lines $l_1$ and $l_2$ have equations $\mathbf{r}=\mathbf{i}+4\mathbf{j}-\mathbf{k}+\lambda (\mathbf{j}-2\mathbf{k})$ and $\mathbf{r}=-3\mathbf{i}+4\mathbf{j}+\mu (\mathbf{i}+2\mathbf{j}+\mathbf{k})$ respectively.

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• Question 7May/June 2023, paper 12Structured14 marks

  The plane ${\mathrm{\Pi }}_1$ has equation $\mathbf{r}=-4\mathbf{j}-3\mathbf{k}+\lambda (\mathbf{i}-\mathbf{j}+\mathbf{k})+\mu (\mathbf{i}+\mathbf{j}-\mathbf{k})$. Obtain an equation of ${\mathrm{\Pi }}_1$ in the form $px+qy+rz=d$.

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• Question 8May/June 2024, paper 23Structured14 marks

  The planes ${\mathrm{\Pi }}_1$ and ${\mathrm{\Pi }}_2$ do not intersect and are both perpendicular to $\mathbf{i}+2\mathbf{j}+3\mathbf{k}$. The line $l$ intersects ${\mathrm{\Pi }}_1$ at the point $(1,6,0)$ and intersects ${\mathrm{\Pi }}_2$ at the point $(3,-6,0)$. Find Cartesian equations of ${\mathrm{\Pi }}_1$ and ${\mathrm{\Pi }}_2$.

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