## Vector Line and Plane Equation

### Vector Line and Plane Equation

We have learned the cartesian form of a line equation: $$\boxed{ \ y \ = \ mx \ + \ c \ }$$.

While it comes in handy in solving one-dimensional (1D) and two-dimensional (2D) problems, it may not be as such when solving three-dimensional problems.

Using the vector form of a line equation and a plane equation helps us to solve 3D problems much easier than using its cartesian form.

Given any two points, A and B, we can draw the vector $${\small \vec{a}}$$ and $${\small \vec{b}}$$ from the origin. Then, the line equation of line AB in the vector form can be written as follows:

$$\hspace{3em} \vec{r} \ = \ \vec{a} \ + \ \lambda(\vec{b} \ – \ \vec{a})$$

$$\\[5pt] {\small \vec{r} \ = \ }$$ position vector
: it represents any points along the line AB

$$\\[5pt] {\small \vec{a} \ \ \textrm{or} \ \ \vec{b} \ = \ }$$ location vector
: it shows the location vector of any one point along the line AB which can be represented by $${\small \vec{a}}$$ or $${\small \vec{b}}$$

$$\\[5pt] {\small \vec{b} \ – \ \vec{a} \ = \ }$$ direction vector
: it gives the direction vector of the line AB

$$\\[5pt] {\small \lambda \ }$$ is a constant value.

For the 2D shape, the vector form of a plane equation is shown below:

$$\hspace{3em} (\vec{r} \ – \ \vec{a}) \cdot \vec{n} \ = \ 0$$

$$\\[5pt] {\small \vec{r} \ = \ }$$ position vector
: it represents any points on the plane

$$\\[5pt] {\small \vec{a} \ = \ }$$ location vector
: it shows the location of a point on the plane which is represented by $${\small \vec{a}}$$

$$\\[5pt] {\small \vec{n} \ = \ }$$ normal vector
: it is the vector that gives perpendicular direction to the plane.

Note that we can find the cartesian form of a plane equation from its vector form,

$$\\[8pt] \hspace{3em} (\vec{r} \ – \ \vec{a}) \cdot \vec{n} \ \hspace{0.7em} \ = \ 0$$
$$\\[8pt] \hspace{3em} \vec{r} \ \cdot \vec{n} \ – \ \vec{a} \ \cdot \vec{n} \ = \ 0$$
$$\\[8pt] \hspace{3em} \vec{r} \ \cdot \vec{n} \hspace{3.3em} \ = \ \vec{a} \ \cdot \vec{n}$$

Let $$\ {\small \vec{r}}=\begin{pmatrix} x \\[1pt] y \\[1pt] z \end{pmatrix} \$$ and $$\ {\small \vec{n}}=\begin{pmatrix} a \\[1pt] b \\[1pt] c \end{pmatrix}$$

with $${\small \ (x,y,z) \ }$$ are the cartesian coordinates of any points on the plane and $${\small \ (a,b,c)} \$$ are the cartesian components of the normal vector $$\ {\small \vec{n} \ }$$. Then the cartesian form is:

$$\hspace{3em} ax \ + \ by \ + \ cz \ = \ d$$

with $$\ {\small d \ = \ \vec{a} \ \cdot \vec{n} \ }$$ (the dot product of vector $$\ {\small \vec{a} \ }$$ and $$\ {\small \vec{n} \ }$$ ).

I have put together some of the questions I received in the comment section below. You can try these questions also to further your understanding on this topic.

To check your answer, you can look through the solutions that I have posted either in Youtube videos or Instagram posts.

You can subscribe, like or follow my youtube channel and IG account. I will keep updating my IG daily post, preferably.

Furthermore, you can find some examples and more practices below! =).

Try some of the examples below and if you need any help, just look at the solution I have written. Cheers ! =) .
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EXAMPLE:

$${\small 1.\enspace}$$ 9709/03/SP/20 – Specimen Paper 2020 Pure Maths 3 No 8
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$$\\[1pt]$$
In the diagram, OABC is a pyramid in which OA = 2 units, OB = 4 units and OC = 2 units. The edge OC is vertical, the base OAB is horizontal and angle $${\small \ AOB \ = \ 90^{\large{\circ}}}$$. Unit vectors i, j and k are parallel to OA, OB and OC respectively. The midpoints of AB and BC are M and N respectively.
$$\\[1pt]$$
$${\small\hspace{1.2em}(\textrm{a}).\hspace{0.8em}}$$ Express the vectors $${\small \ \overrightarrow{ON} \ }$$ and $${\small \ \overrightarrow{CM} \ }$$ in terms of i, j and k.
$$\\[1pt]$$
$${\small\hspace{1.2em}(\textrm{b}).\hspace{0.8em}}$$ Calculate the angle between the directions of $${\small \ \overrightarrow{ON} \ }$$ and $${\small \ \overrightarrow{CM} \ }$$.
$$\\[1pt]$$
$${\small\hspace{1.2em}(\textrm{c}).\hspace{0.8em}}$$ Show that the length of the perpendicular from M to ON is $${\small \ {\large \frac{3}{5} } \sqrt 5 }$$.

$$\\[1pt]$$
$${\small 2.\enspace}$$ 9709/11/O/N/16 – Paper 11 Oct Nov 2020 Pure Maths 1 No 9
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$$\\[1pt]$$
The diagram shows a cuboid OABCDEFG with a horizontal base OABC in which OA = 4 cm and AB = 15 cm. The height OD of the cuboid is 2 cm. The point X on AB is such that AX = 5 cm and the point P on DG is such that DP = p cm, where p is a constant. Unit vectors i, j and k are parallel to OA, OC and OD respectively.
$$\\[1pt]$$
$${\small\hspace{1.2em}(\textrm{i}).\hspace{0.7em}}$$ Find the possible values of p such that angle $${\small \ OPX \ = \ 90^{\large{\circ}}}$$.
$$\\[1pt]$$
$${\small\hspace{1.2em}(\textrm{ii}).\hspace{0.7em}}$$ For the case where p = 9, find the unit vector in the direction of $${\small \ \overrightarrow{XP} }$$.
$$\\[1pt]$$
$${\small\hspace{1.2em}(\textrm{iii}).\hspace{0.5em}}$$ A point Q lies on the face CBFG and is such that XQ is parallel to AG. Find $${\small \ \overrightarrow{XQ} }$$.

$$\\[1pt]$$
$${\small 3.\enspace}$$ The points A and B have position vectors i + 2jk and 3i + j + k respectively. The line l has equation r = 2i + j + k + $${\small \mu}$$(i + j + 2k).
$$\\[1pt]$$
$${\small\hspace{1.2em}\textrm{(i)}.\hspace{0.8em}}$$ Show that l does not intersect the line passing through A and B.
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$${\small\hspace{1.2em}\textrm{(ii)}.\hspace{0.8em}}$$ The plane m is perpendicular to AB and passes through the mid-point of AB. The plane m intersects the line l at the point P. Find the equation of m and the position vector of P.

$$\\[1pt]$$
$${\small 4.\enspace}$$

$$\\[1pt]$$
The diagram shows a set of rectangular axes Ox, Oy and Oz, and four points A, B, C and D with position vectors $${\small \overrightarrow{OA} \ = \ 3\textbf{i} }$$, $${\small \overrightarrow{OB} \ = \ 3\textbf{i} \ + \ 4\textbf{j} \ }$$, $${\small \overrightarrow{OC} \ = \ \textbf{i} \ + \ 3\textbf{j} \ }$$ and $${\small \overrightarrow{OD} \ = \ 2\textbf{i} \ + \ 3\textbf{j} \ + \ 5\textbf{k} \ }$$.
$$\\[1pt]$$
$${\small\hspace{1.2em}\textrm{(i)}.\hspace{0.8em}}$$ Find the equation of the plane BCD, giving your answer in the form $${\small ax + by + cz = d}$$.
$$\\[1pt]$$
$${\small\hspace{1.2em}\textrm{(ii)}.\hspace{0.8em}}$$ Calculate the acute angle between the planes BCD and OABC .

$$\\[1pt]$$
$${\small 5.\enspace}$$ The line l has equation r = i + 2j + 3k + $${\small \mu}$$(2ij – 2k).
$$\\[1pt]$$
$${\small\hspace{1.2em}\textrm{(i)}.\hspace{0.8em}}$$ The point P has position vector 4i + 2j – 3k. Find the length of the perpendicular from P to l.
$$\\[1pt]$$
$${\small\hspace{1.2em}\textrm{(ii)}.\hspace{0.8em}}$$ It is given that l lies in the plane $${\small ax + by + 2z = 13}$$, where a and b are constants. Find the values of a and b.

$$\\[1pt]$$
$${\small 6.\enspace}$$ Two planes have equations $${\small 2x + 3y \ – \ z \ = \ 1 \ }$$ and $$\ {\small x \ – \ 2y + z \ = \ 3}$$.
$$\\[1pt]$$
$${\small\hspace{1.2em}\textrm{(i)}.\hspace{0.8em}}$$ Find the acute angle between the planes.
$$\\[1pt]$$
$${\small\hspace{1.2em}\textrm{(ii)}.\hspace{0.8em}}$$ Find a vector equation for the line of intersection of the planes.

$$\\[1pt]$$
$${\small 7.\enspace}$$ The planes m and n have equations $${\small 3x + y \ – \ 2z \ = \ 10}$$ and $${\small x \ – \ 2y + 2z \ = \ 5}$$ respectively. The line l has equation r = 4i + 2j + k + $${\small \lambda}$$(i + j + 2k).
$$\\[1pt]$$
$${\small\hspace{1.2em}\textrm{(i)}.\hspace{0.8em}}$$ Show that l is parallel to m.
$$\\[1pt]$$
$${\small\hspace{1.2em}\textrm{(ii)}.\hspace{0.8em}}$$ Calculate the acute angle between the planes m and n.
$$\\[1pt]$$
$${\small\hspace{1.2em}\textrm{(iii)}.\hspace{0.8em}}$$ A point P lies on the line l. The perpendicular distance P from the plane l is equal to 2. Find the position vectors of the two possible positions of P.

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$${\small 8.\enspace}$$ The line l has equation r = 5i – 3jk + $${\small \lambda}$$(i – 2j + k). The plane p has equation (ri – 2j) . (3i + j + k) = 0. The line l intersects the plane p at the point A.
$$\\[1pt]$$
$${\small\hspace{1.2em}\textrm{(i)}.\hspace{0.8em}}$$ Find the position vector of A.
$$\\[1pt]$$
$${\small\hspace{1.2em}\textrm{(ii)}.\hspace{0.8em}}$$ Calculate the acute angle between l and p.
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$${\small\hspace{1.2em}\textrm{(iii)}.\hspace{0.8em}}$$ Find the equation of the line which lies in p and intersects l at right angles.

$$\\[1pt]$$
$${\small 9.\enspace}$$ 9709/33/M/J/20 – Paper 33 June 2020 Pure Maths 3 No 8
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Relative to the origin $$O$$, the points $$A$$, $$B$$ and $$D$$ have position vectors given by
$$\\[1pt]$$
$${\small \ \overrightarrow{OA} = \textbf{i} \ + \ 2\textbf{j} \ + \ \textbf{k} \ }$$, $${\small \ \overrightarrow{OB} = 2\textbf{i} \ + \ 5\textbf{j} \ + \ 3\textbf{k} \ }$$ and $${\small \ \overrightarrow{OD} = 3\textbf{i} \ + \ 2\textbf{k} }$$.
$$\\[1pt]$$
A fourth point $$C$$ is such that $$ABCD$$ is a parallelogram.
$$\\[1pt]$$
$${\small (\textrm{a}).\hspace{0.8em}}$$ Find the position vector of $$C$$ and verify that the parallelogram is not a rhombus.
$$\\[1pt]$$
$${\small (\textrm{b}).\hspace{0.8em}}$$ Find angle $$BAD$$, giving your answer in degrees.
$$\\[1pt]$$
$${\small (\textrm{c}).\hspace{0.8em}}$$ Find the area of the parallelogram correct to 3 significant figures.

$$\\[1pt]$$
$${\small 10.\enspace}$$ 9709/32/M/J/20 – Paper 32 June 2020 Pure Maths 3 No 10
$$\\[1pt]$$
With respect to the origin $$O$$, the points $$A$$ and $$B$$ have position vectors given by $${\small \ \overrightarrow{OA} = 6\textbf{i} \ + \ 2\textbf{j} \ }$$ and $${\small \ \overrightarrow{OB} = 2\textbf{i} \ + \ 2\textbf{j} \ + \ 3\textbf{k} }$$. The midpoint of $$OA$$ is $$M$$. The point $$N$$ lying on $$AB$$, between $$A$$ and $$B$$, is such that $$AN = 2NB$$.
$$\\[1pt]$$
$${\small (\textrm{a}).\hspace{0.8em}}$$ Find a vector equation for the line through $$M$$ and $$N$$.
$$\\[1pt]$$
The line through $$M$$ and $$N$$ intersects the line through $$O$$ and $$B$$ at the point $$P$$.
$$\\[1pt]$$
$${\small (\textrm{b}).\hspace{0.8em}}$$ Find the position vector of $$P$$.
$$\\[1pt]$$
$${\small (\textrm{c}).\hspace{0.8em}}$$ Calculate angle $$OPM$$, giving your answer in degrees.

$$\\[1pt]$$
$${\small 11.\enspace}$$ 9709/31/M/J/20 – Paper 31 June 2020 Pure Maths 3 No 9
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With respect to the origin $$O$$, the vertices of a triangle $$ABC$$ have position vectors
$$\\[1pt]$$
$${\small \ \overrightarrow{OA} = 2\textbf{i} \ + \ 5\textbf{k} \ }$$, $${\small \ \overrightarrow{OB} = 3\textbf{i} \ + \ 2\textbf{j} \ + \ 3\textbf{k} \ }$$ and $${\small \ \overrightarrow{OC} = \textbf{i} \ + \ \textbf{j} \ + \ \textbf{k} }$$.
$$\\[1pt]$$
$${\small (\textrm{a}).\hspace{0.8em}}$$ Using a scalar product, show that angle $$ABC$$ is a right angle.
$$\\[1pt]$$
$${\small (\textrm{b}).\hspace{0.8em}}$$ Show that triangle $$ABC$$ is isosceles.
$$\\[1pt]$$
$${\small (\textrm{c}).\hspace{0.8em}}$$ Find the exact length of the perpendicular from $$O$$ to the line through $$B$$ and $$C$$.

$$\\[1pt]$$
$${\small 12.\enspace}$$ 9709/32/F/M/21 – Paper 32 March 2021 Pure Maths 3 No 7
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Two lines have equations $$\ {\small \vec{r}}=\begin{pmatrix} 1 \\[1pt] 3 \\[1pt] 2 \end{pmatrix} + s \begin{pmatrix} 2 \\[1pt] -1 \\[1pt] 3 \end{pmatrix}$$ and $$\ {\small \vec{r}}=\begin{pmatrix} 2 \\[1pt] 1 \\[1pt] 4 \end{pmatrix} + t \begin{pmatrix} 1 \\[1pt] -1 \\[1pt] 4 \end{pmatrix}$$
$$\\[1pt]$$
$${\small (\textrm{a}).\hspace{0.8em}}$$ Show that the lines are skew.
$$\\[1pt]$$
$${\small (\textrm{b}).\hspace{0.8em}}$$ Find the acute angle between the directions of the two lines.

$$\\[1pt]$$
$${\small 13.\enspace}$$ 9709/33/M/J/21 – Paper 33 June 2021 Pure Maths 3 No 9
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The quadrilateral $$ABCD$$ is a trapezium in which $$AB$$ and $$DC$$ are parallel. With respect to the origin $$O$$, the position vectors of $$A$$, $$B$$ and $$C$$ are given by
$$\\[1pt]$$
$${\small \ \overrightarrow{OA} = -\textbf{i} \ + \ 2\textbf{j} \ + \ 3\textbf{k} \ }$$, $${\small \ \overrightarrow{OB} = \textbf{i} \ + \ 3\textbf{j} \ + \ \textbf{k} \ }$$ and $${\small \ \overrightarrow{OC} = 2\textbf{i} \ + \ 2\textbf{j} \ – \ 3\textbf{k} }$$.
$$\\[1pt]$$
$${\small (\textrm{a}).\hspace{0.8em}}$$ Given that $${\small \ \overrightarrow{DC} = 3 \overrightarrow{AB} }$$, find the position vector of $$D$$.
$$\\[1pt]$$
$${\small (\textrm{b}).\hspace{0.8em}}$$ State a vector equation for the line through $$A$$ and $$B$$.
$$\\[1pt]$$
$${\small (\textrm{c}).\hspace{0.8em}}$$ Find the distance between the parallel sides and hence find the area of the trapezium.

$$\\[1pt]$$
$${\small 14.\enspace}$$ 9709/32/M/J/21 – Paper 32 June 2021 Pure Maths 3 No 11
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With respect to the origin $$O$$, the points $$A$$ and $$B$$ have position vectors given by
$$\\[1pt]$$
$${\small \ \overrightarrow{OA} = 2\textbf{i} \ – \ \textbf{j} \ }$$ and $${\small \ \overrightarrow{OB} = \textbf{j} \ – \ 2\textbf{k} }$$.
$$\\[1pt]$$
$${\small (\textrm{a}).\hspace{0.8em}}$$ Show that $$OA = OB$$ and use a scalar product to calculate angle $$AOB$$ in degrees.
$$\\[1pt]$$
The midpoint of $$AB$$ is $$M$$. The point $$P$$ on the line through $$O$$ and $$M$$ is such that $$PA : OA = \sqrt{7} : 1$$.
$$\\[1pt]$$
$${\small (\textrm{b}).\hspace{0.8em}}$$ Find the possible position vectors of $$P$$.

$$\\[1pt]$$
$${\small 15.\enspace}$$ 9709/31/M/J/21 – Paper 31 June 2021 Pure Maths 3 No 8
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With respect to the origin $$O$$, the points $$A$$ and $$B$$ have position vectors given by
$$\\[1pt]$$
$$\ {\small \overrightarrow{OA}}=\begin{pmatrix} 1 \\[1pt] 2 \\[1pt] 1 \end{pmatrix}$$ and $$\ {\small \overrightarrow{OB}}=\begin{pmatrix} 3 \\[1pt] 1 \\[1pt] -2 \end{pmatrix}$$. The line $$l$$ has equation $$\ {\small \vec{r}}=\begin{pmatrix} 2 \\[1pt] 3 \\[1pt] 1 \end{pmatrix} + \lambda \begin{pmatrix} 1 \\[1pt] -2 \\[1pt] 1 \end{pmatrix}$$.
$$\\[1pt]$$
$${\small (\textrm{a}).\hspace{0.8em}}$$ Find the acute angle between the directions of $$AB$$ and $$l$$.
$$\\[1pt]$$
$${\small (\textrm{b}).\hspace{0.8em}}$$ Find the position vector of the point $$P$$ on $$l$$ such that $$AP = BP$$.

$$\\[1pt]$$
$${\small 16.\enspace}$$ 9709/32/F/M/22 – Paper 32 March 2022 Pure Maths 3 No 10
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The points $$A$$ and $$B$$ have position vectors $${\small \ 2\textbf{i} \ + \ \textbf{j} \ + \ \textbf{k} \ }$$ and $${\small \ \textbf{i} \ – \ 2\textbf{j} \ + \ 2\textbf{k} \ }$$ respectively.
$$\\[1pt]$$
The line $$l$$ has vector equation $${\small \ \vec{r} \ = \ \textbf{i} \ + \ 2\textbf{j} \ – \ 3\textbf{k} \ + \ \mu ( \textbf{i} \ – \ 3\textbf{j} \ – \ 2\textbf{k} ) }$$.
$$\\[1pt]$$
$${\small (\textrm{a}).\hspace{0.8em}}$$ Find a vector equation for the line through $$A$$ and $$B$$.
$$\\[1pt]$$
$${\small (\textrm{b}).\hspace{0.8em}}$$ Find the acute angle between the directions of $$AB$$ and $$l$$, giving your answer in degrees.
$$\\[1pt]$$
$${\small (\textrm{c}).\hspace{0.8em}}$$ Show that the line through $$A$$ and $$B$$ does not intersect the line $$l$$.

$$\\[1pt]$$
$${\small 17.\enspace}$$ 9709/12/O/N/19 – Paper 12 Oct Nov 2019 Pure Maths 1 No 7
$$\\[1pt]$$

$$\\[1pt]$$
The base OABC and the upper surface DEFG are identical horizontal rectangles. The parallelograms OAED and CBFG both lie in vertical planes. Points P and Q are the mid-points of OD and GF respectively. Unit vectors i and j are parallel to $${\small \ \overrightarrow{OA} \ }$$ and $${\small \ \overrightarrow{OC} \ }$$ respectively and the unit vector k is vertically upwards. The position vectors of A, C and D are given by $${\small \ \overrightarrow{OA} \ = \ 6\textbf{i} }$$, $${\small \ \overrightarrow{OC} \ = \ 8\textbf{j} }$$ and $${\small \ \overrightarrow{OD} \ = \ 2\textbf{i} \ + \ 10\textbf{k} }$$.
$$\\[1pt]$$
$${\small\hspace{1.2em}(\textrm{i}).\hspace{0.7em}}$$ Express each of the vectors $${\small \ \overrightarrow{PB} \ }$$ and $${\small \ \overrightarrow{PQ} \ }$$ in terms of i, j and k.
$$\\[1pt]$$
$${\small\hspace{1.2em}(\textrm{ii}).\hspace{0.7em}}$$ Determine whether P is nearer to Q or to B.
$$\\[1pt]$$
$${\small\hspace{1.2em}(\textrm{iii}).\hspace{0.5em}}$$ Use a scalar product to find angle BPQ.

$$\\[1pt]$$

$$\\[1pt]$$

PRACTICE MORE WITH THESE QUESTIONS BELOW!

$${\small 1.\enspace}$$ The points A and B have position vectors, relative to the origin O, given by $${\small \overrightarrow{OA} \ = \ \textbf{i} \ + \ \textbf{j} \ + \ \textbf{k} }$$ and $${\small \overrightarrow{OB} \ = \ 2\textbf{i} \ + \ 3\ \textbf{k} }$$. The line l has vector equation $${\small \vec{r} = 2\textbf{i} \ – \ 2\textbf{j} \ – \ \textbf{k} + \mu(-\textbf{i} + 2\textbf{j} + \textbf{k} ) }$$.
$$\\[1pt]$$
$${\small\hspace{2.8em}\textrm{(i)}.\hspace{0.7em}}$$ Show that the line passing through A and B does not intersect l.
$$\\[1pt]$$
$${\small\hspace{2.8em}\textrm{(ii)}.\hspace{0.7em}}$$ Show that the length of the perpendicular from A to l is $${\small {\large\frac{1}{\sqrt{2}} }}$$.
$$\\[1pt]$$

$${\small 2. \enspace}$$ The point P has position vector $${\small 3\textbf{i} \ – \ 2\textbf{j} \ + \ \textbf{k} }$$. The line l has equation $${\small \vec{r} = 4\textbf{i} + 2\textbf{j} + 5\textbf{k} + \mu(\textbf{i} + 2\textbf{j} + 3\textbf{k} ) }$$.
$$\\[1pt]$$
$${\small\hspace{2.8em}\textrm{(i)}.\hspace{0.7em}}$$ Find the length of the perpendicular from P to l, giving your answer correct to 3 significant figures.
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$${\small\hspace{2.8em}\textrm{(ii)}.\hspace{0.7em}}$$ Find the equation of the plane containing l and P, giving your answer in the form $${\small ax + by + cz = d}$$.
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$${\small 3. \enspace}$$ Two lines l and m have equations $${\small \vec{r} = 2\textbf{i} \ – \textbf{j} + \textbf{k} + s(2\textbf{i} + 3\textbf{j} \ – \textbf{k} ) }$$ and $${\small \vec{r} = \textbf{i} + 3\textbf{j} + 4\textbf{k} + t(\textbf{i} + 2\textbf{j} + \textbf{k} ) }$$ respectively.
$$\\[1pt]$$
$${\small\hspace{2.8em}\textrm{(i)}.\hspace{0.7em}}$$ Show that the lines are skew.
$$\\[1pt]$$
$${\small\hspace{2.8em}\textrm{(ii)}.\hspace{0.7em}}$$ A plane p is parallel to the lines l and m. Find a vector that is normal to p.
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$${\small\hspace{2.8em}\textrm{(iii)}.\hspace{0.5em}}$$ Given that p is equidistant from the lines l and m, find the equation of p. Give your answer in
the form $${\small ax + by + cz = d}$$.
$$\\[1pt]$$

$${\small 4. \enspace}$$ The line l has equation $${\small \vec{r} = 4\textbf{i} + 3\textbf{j} \ – \textbf{k} + \mu(\textbf{i} + 2\textbf{j} \ – 2\textbf{k} ) }$$. The plane p has equation $${\small 2x \ – 3y \ – z = 4}$$.
$$\\[1pt]$$
$${\small\hspace{2.8em}\textrm{(i)}.\hspace{0.7em}}$$ Find the position vector of the point of intersection of l and p.
$$\\[1pt]$$
$${\small\hspace{2.8em}\textrm{(ii)}.\hspace{0.7em}}$$ Find the acute angle between l and p.
$$\\[1pt]$$
$${\small\hspace{2.8em}\textrm{(iii)}.\hspace{0.5em}}$$ A second plane q is parallel to l, perpendicular to p and contains the point with position vector
4jk. Find the equation of q, giving your answer in the form $${\small ax + by + cz = d}$$.
$$\\[1pt]$$

$${\small 5. \enspace}$$ Two planes p and q have equations $${\small x + y + 3z = 8}$$ and $${\small 2x \ – 2y + z = 3}$$ respectively.
$$\\[1pt]$$
$${\small\hspace{2.8em}\textrm{(i)}.\hspace{0.7em}}$$ Calculate the acute angle between the planes p and q.
$$\\[1pt]$$
$${\small\hspace{2.8em}\textrm{(ii)}.\hspace{0.7em}}$$ The point A on the line of intersection of p and q has y-coordinate equal to 2. Find the equation of the plane which contains the point A and is perpendicular to both the planes p and q. Give your answer in the form $${\small ax + by + cz = d}$$.
$$\\[1pt]$$

$${\small 6. \enspace}$$ The equations of two lines l and m are $${\small \vec{r} = 3\textbf{i} \ – \textbf{j} \ – 2\textbf{k} + \lambda(\ -\textbf{i} + \textbf{j} + 4\textbf{k} ) }$$ and $${\small \vec{r} = 4\textbf{i} + 4\textbf{j} \ – 3\textbf{k} + \mu(2\textbf{i} + \textbf{j} \ – 2\textbf{k} ) }$$ respectively.
$$\\[1pt]$$
$${\small\hspace{2.8em}\textrm{(i)}.\hspace{0.7em}}$$ Show that the lines do not intersect.
$$\\[1pt]$$
$${\small\hspace{2.8em}\textrm{(ii)}.\hspace{0.7em}}$$ ) Calculate the acute angle between the directions of the lines.
$$\\[1pt]$$
$${\small\hspace{2.8em}\textrm{(iii)}.\hspace{0.5em}}$$ Find the equation of the plane which passes through the point (3, −2, −1) and which is parallel to both l and m. Give your answer in the form $${\small ax + by + cz = d}$$.
$$\\[1pt]$$

$${\small 7. \enspace}$$ The points A and B have position vectors, relative to the origin O, given by $${\small \overrightarrow{OA} \ = \ \textbf{i} \ + \ 2\textbf{j} \ + \ 3\textbf{k} }$$ and $${\small \overrightarrow{OB} \ = \ 2\textbf{i} \ + \ \textbf{j} \ + \ 3\ \textbf{k} }$$. The line l has vector equation $${\small \vec{r} = (1 \ – \ 2t)\textbf{i} + (5 \ + \ t)\textbf{j} \ + (2 \ – \ t)\textbf{k} }$$.
$$\\[1pt]$$
$${\small\hspace{2.8em}\textrm{(i)}.\hspace{0.7em}}$$ Show that l does not intersect the line passing through A and B.
$$\\[1pt]$$
$${\small\hspace{2.8em}\textrm{(ii)}.\hspace{0.7em}}$$ ) The point P lies on l and is such that angle PAB is equal to $${\small 60^{\large{\circ}} }$$. Given that the position vector of P is $${\small (1 \ – \ 2t)\textbf{i} + (5 \ + \ t)\textbf{j} \ + (2 \ – \ t)\textbf{k} }$$, show that $${\small 3t^2 \ + \ 7t \ + \ 2 \ = \ 0 }$$. Hence find the only possible position vector of P.
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$${\small 8. \enspace}$$ The plane p has equation $${\small 3x + 2y + 4z = 13}$$. A second plane q is perpendicular to p and has equation $${\small ax + y + z = 4}$$, where a is a constant.
$$\\[1pt]$$
$${\small\hspace{2.8em}\textrm{(i)}.\hspace{0.7em}}$$ Find the value of a.
$$\\[1pt]$$
$${\small\hspace{2.8em}\textrm{(ii)}.\hspace{0.7em}}$$ The line with equation $${\small \vec{r} = \textbf{j} \ – \textbf{k} + \lambda( \textbf{i} + 2\textbf{j} + 2\textbf{k} ) }$$ meets the plane p at the point A and the plane q at the point B. Find the length of AB.
$$\\[1pt]$$

$${\small 9. \enspace}$$ The lines $${\small l_{1}}$$ and $${\small l_{2}}$$ have equations $${\small \vec{r} = \textbf{i} + 2\textbf{j} + 3\textbf{k} + \lambda(a\textbf{i} + 4\textbf{j} + 3 \textbf{k} ) }$$ and $${\small \vec{r} = 4\textbf{i} \ – \textbf{k} + \mu(2\textbf{i} + 4\textbf{j} + b\textbf{k} ) }$$ respectively. Given that $${\small l_{1}}$$ and $${\small l_{2}}$$ are parallel:
$$\\[1pt]$$
$${\small\hspace{2.8em}\textrm{(i)}.\hspace{0.7em}}$$ Write down the values of a and b.
$$\\[1pt]$$
$${\small\hspace{2.8em}\textrm{(ii)}.\hspace{0.7em}}$$ Find the shortest distance d between $${\small l_{1}}$$ and $${\small l_{2}}$$.
$$\\[1pt]$$
$${\small\hspace{2.8em}\textrm{(iii)}.\hspace{0.5em}}$$ Find a vector equation of the plane p containing $${\small l_{1}}$$ and $${\small l_{2}}$$.
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$${\small 10.\enspace}$$ Two lines $${\small l_{1}}$$ and $${\small l_{2}}$$ have equations $${\small \vec{r} = \ -8\textbf{i} + 12\textbf{j} + 16\textbf{k} + \lambda(\textbf{i} + 7\textbf{j} \ – 2 \textbf{k} ) }$$ and $${\small \vec{r} = 4\textbf{i} + 6 \textbf{j} \ – 8\textbf{k} + \mu(2\textbf{i} \ – \textbf{j} + 2\textbf{k} ) }$$ respectively.
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$${\small\hspace{2.8em}\textrm{(i)}.\hspace{0.7em}}$$ Show that $${\small l_{1}}$$ and $${\small l_{2}}$$ are skew lines.
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$${\small\hspace{2.8em}\textrm{(ii)}.\hspace{0.7em}}$$ The points P and Q lie on $${\small l_{1}}$$ and $${\small l_{2}}$$ respectively such that PQ is perpendicular to both $${\small l_{1}}$$ and $${\small l_{2}}$$. Show that $${\small \overrightarrow{PQ} \ = \ 16\textbf{i} \ – \ 8\textbf{j} \ – \ 20\textbf{k} }$$.
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