4.04a Line equations: 2D and 3D, cartesian and vector forms

1. Relative to a fixed origin \(O\), the point \(A\) has position vector \(( 10 \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } )\), and the point \(B\) has position vector \(( 8 \mathbf { i } + 3 \mathbf { j } + 4 \mathbf { k } )\).

The line \(l\) passes through the points \(A\) and \(B\).

1. Find the vector \(\overrightarrow { A B }\).
2. Find a vector equation for the line \(l\). The point \(C\) has position vector \(( 3 \mathbf { i } + 12 \mathbf { j } + 3 \mathbf { k } )\).
   The point \(P\) lies on \(l\). Given that the vector \(\overrightarrow { C P }\) is perpendicular to \(l\),
3. find the position vector of the point \(P\).

6. Relative to a fixed origin \(O\), the point \(A\) has position vector \(21 \mathbf { i } - 17 \mathbf { j } + 6 \mathbf { k }\) and the point \(B\) has position vector \(25 \mathbf { i } - 14 \mathbf { j } + 18 \mathbf { k }\). The line \(l\) has vector equation $$\mathbf { r } = \left( \begin{array} { r } a \\ b \\ 10 \end{array} \right) + \lambda \left( \begin{array} { r } 6 \\ c \\ - 1 \end{array} \right)$$ where \(a , b\) and \(c\) are constants and \(\lambda\) is a parameter.
Given that the point \(A\) lies on the line \(l\),

1. find the value of \(a\). Given also that the vector \(\overrightarrow { A B }\) is perpendicular to \(l\),
2. find the values of \(b\) and \(c\),
3. find the distance \(A B\). The image of the point \(B\) after reflection in the line \(l\) is the point \(B ^ { \prime }\).
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6. With respect to a fixed origin \(O\), the lines \(l _ { 1 }\) and \(l _ { 2 }\) are given by the equations $$l _ { 1 } : \mathbf { r } = \left( \begin{array} { r } 4 \\ 28 \\ 4 \end{array} \right) + \lambda \left( \begin{array} { r } - 1 \\ - 5 \\ 1 \end{array} \right) , \quad l _ { 2 } : \mathbf { r } = \left( \begin{array} { l } 5 \\ 3 \\ 1 \end{array} \right) + \mu \left( \begin{array} { r } 3 \\ 0 \\ - 4 \end{array} \right)$$ where \(\lambda\) and \(\mu\) are scalar parameters. The lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect at the point \(X\).

1. Find the coordinates of the point \(X\).
2. Find the size of the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\), giving your answer in degrees to 2 decimal places. The point \(A\) lies on \(l _ { 1 }\) and has position vector \(\left( \begin{array} { r } 2 \\ 18 \\ 6 \end{array} \right)\)
3. Find the distance \(A X\), giving your answer as a surd in its simplest form. The point \(Y\) lies on \(l _ { 2 }\). Given that the vector \(\overrightarrow { Y A }\) is perpendicular to the line \(l _ { 1 }\)
4. find the distance \(Y A\), giving your answer to one decimal place. The point \(B\) lies on \(l _ { 1 }\) where \(| \overrightarrow { A X } | = 2 | \overrightarrow { A B } |\).
5. Find the two possible position vectors of \(B\).

5. The vector equations of two straight lines are $$\begin{aligned} & \mathbf { r } = 5 \mathbf { i } + 3 \mathbf { j } - 2 \mathbf { k } + \lambda ( \mathbf { i } - 2 \mathbf { j } + 2 \mathbf { k } ) \quad \text { and } \\ & \mathbf { r } = 2 \mathbf { i } - 11 \mathbf { j } + a \mathbf { k } + \mu ( - 3 \mathbf { i } - 4 \mathbf { j } + 5 \mathbf { k } ) . \end{aligned}$$ Given that the two lines intersect, find

1. the coordinates of the point of intersection,
2. the value of the constant \(a\),
3. the acute angle between the two lines.

1. Relative to a fixed origin \(O\), the line \(l\) has equation

$$\mathbf { r } = \left( \begin{array} { r } 1 \\ - 10 \\ - 9 \end{array} \right) + \lambda \left( \begin{array} { l } 4 \\ 4 \\ 2 \end{array} \right) \quad \text { where } \lambda \text { is a scalar parameter }$$ Given that \(\overrightarrow { O A }\) is a unit vector parallel to \(l\),

1. find \(\overrightarrow { O A }\) The point \(X\) lies on \(l\).
   Given that \(X\) is the point on \(l\) that is closest to the origin,
2. find the coordinates of \(X\). The points \(O , X\) and \(A\) form the triangle \(O X A\).
3. Find the exact area of triangle \(O X A\).

1. Relative to a fixed origin \(O\),

• the point \(A\) has position vector \(\quad \mathbf { i } - 4 \mathbf { j } + 3 \mathbf { k }\)
• the point \(B\) has position vector \(5 \mathbf { i } + 3 \mathbf { j } - 2 \mathbf { k }\)
• the point \(C\) has position vector \(3 \mathbf { i } + p \mathbf { j } - \mathbf { k }\) where \(p\) is a constant.
  The line \(l\) passes through \(A\) and \(B\).
  1. Find a vector equation for the line \(l\)

Given that \(\overrightarrow { A C }\) is perpendicular to \(l\)

find the value of \(p\)

Hence find the area of triangle \(A B C\), giving your answer as a surd in simplest form.

1. Relative to a fixed origin \(O\),

• the point \(A\) has position vector \(4 \mathbf { i } + 8 \mathbf { j } + \mathbf { k }\)
• the point \(B\) has position vector \(5 \mathbf { i } + 6 \mathbf { j } + 3 \mathbf { k }\)
• the point \(P\) has position vector \(2 \mathbf { i } - 2 \mathbf { j } + \mathbf { k }\)

The straight line \(l\) passes through \(A\) and \(B\).

1. Find a vector equation for \(l\). The point \(C\) lies on \(l\) so that \(P C\) is perpendicular to \(l\).
2. Find the coordinates of \(C\). The point \(P ^ { \prime }\) is the reflection of \(P\) in the line \(l\).
3. Find the coordinates of \(P ^ { \prime }\)
4. Hence find \(\left| \overrightarrow { P P ^ { \prime } } \right|\), giving your answer as a simplified surd.

1. The line \(l _ { 1 }\) has equation \(\mathbf { r } = \left( \begin{array} { r } 2 \\ 3 \\ - 7 \end{array} \right) + \lambda \left( \begin{array} { l } 1 \\ 2 \\ 2 \end{array} \right)\) where \(\lambda\) is a scalar parameter.

The line \(l _ { 2 }\) has equation \(\mathbf { r } = \left( \begin{array} { r } 2 \\ 3 \\ - 7 \end{array} \right) + \mu \left( \begin{array} { r } 4 \\ - 1 \\ 8 \end{array} \right)\) where \(\mu\) is a scalar parameter.
Given that \(l _ { 1 }\) and \(l _ { 2 }\) meet at the point \(P\)

1. state the coordinates of \(P\) Given that the angle between lines \(l _ { 1 }\) and \(l _ { 2 }\) is \(\theta\)
2. find the value of \(\cos \theta\), giving the answer as a fully simplified fraction. The point \(Q\) lies on \(l _ { 1 }\) where \(\lambda = 6\) Given that point \(R\) lies on \(l _ { 2 }\) such that triangle \(Q P R\) is an isosceles triangle with \(P Q = P R\)
3. find the exact area of triangle \(Q P R\)
4. find the coordinates of the possible positions of point \(R\)

1. With respect to a fixed origin \(O\), the line \(l _ { 1 }\) is given by the equation

$$\mathbf { r } = \left( \begin{array} { r } 8 \\ 1 \\ - 3 \end{array} \right) + \mu \left( \begin{array} { r } - 5 \\ 4 \\ 3 \end{array} \right)$$ where \(\mu\) is a scalar parameter.
The point \(A\) lies on \(l _ { 1 }\) where \(\mu = 1\)

1. Find the coordinates of \(A\). The point \(P\) has position vector \(\left( \begin{array} { l } 1 \\ 5 \\ 2 \end{array} \right)\) The line \(l _ { 2 }\) passes through the point \(P\) and is parallel to the line \(l _ { 1 }\)
2. Write down a vector equation for the line \(l _ { 2 }\)
3. Find the exact value of the distance \(A P\). Give your answer in the form \(k \sqrt { 2 }\), where \(k\) is a constant to be found. The acute angle between \(A P\) and \(l _ { 2 }\) is \(\theta\)
4. Find the value of \(\cos \theta\) A point \(E\) lies on the line \(l _ { 2 }\) Given that \(A P = P E\),
5. find the area of triangle \(A P E\),
6. find the coordinates of the two possible positions of \(E\).

7. The line \(l _ { 1 }\) has equation $$\frac { x - 3 } { 4 } = \frac { y - 5 } { - 2 } = \frac { z - 4 } { 7 }$$ The plane \(\Pi\) has equation $$2 x + 4 y - z = 1$$ The line \(l _ { 1 }\) intersects the plane \(\Pi\) at the point \(P\)

1. Determine the coordinates of \(P\) The acute angle between \(l _ { 1 }\) and \(\Pi\) is \(\theta\) degrees.
2. Determine, to one decimal place, the value of \(\theta\) The line \(l _ { 2 }\) lies in \(\Pi\) and passes through \(P\) Given that the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\) is also \(\theta\) degrees,
3. determine a vector equation for \(l _ { 2 }\)

2. $$\mathbf { T } = \left( \begin{array} { l l l } 2 & 3 & 7 \\ 3 & 2 & 6 \\ a & 4 & b \end{array} \right) \quad \mathbf { U } = \left( \begin{array} { r r r } 6 & - 1 & - 4 \\ 15 & c & - 9 \\ - 8 & a & 5 \end{array} \right)$$ where \(a\), \(b\) and \(c\) are constants.
Given that \(\mathbf { T U } = \mathbf { I }\)

1. determine the value of \(a\), the value of \(b\) and the value of \(c\) The transformation represented by the matrix \(\mathbf { T }\) transforms the line \(l _ { 1 }\) to the line \(l _ { 2 }\) Given that \(l _ { 2 }\) has equation $$\frac { x - 1 } { 3 } = \frac { y } { - 4 } = z + 2$$
2. determine a Cartesian equation for \(l _ { 1 }\)

8. The line \(l\) has equation $$\mathbf { r } = ( 2 \mathbf { i } + \mathbf { j } - 2 \mathbf { k } ) + \lambda ( 3 \mathbf { i } + 2 \mathbf { j } + \mathbf { k } ) , \text { where } \lambda \text { is a scalar parameter, }$$ and the plane \(\Pi\) has equation $$\mathbf { r } . ( \mathbf { i } + \mathbf { j } - 2 \mathbf { k } ) = 19$$

1. Find the coordinates of the point of intersection of \(l\) and \(\Pi\). The perpendicular to \(\Pi\) from the point \(A ( 2,1 , - 2 )\) meets \(\Pi\) at the point \(B\).
2. Verify that the coordinates of \(B\) are \(( 4,3 , - 6 )\). The point \(A ( 2,1 , - 2 )\) is reflected in the plane \(\Pi\) to give the image point \(A ^ { \prime }\).
3. Find the coordinates of the point \(A ^ { \prime }\).
4. Find an equation for the line obtained by reflecting the line \(l\) in the plane \(\Pi\), giving your answer in the form $$\mathbf { r } \times \mathbf { a } = \mathbf { b } ,$$ where \(\mathbf { a }\) and \(\mathbf { b }\) are vectors to be found.

7. The plane \(\Pi _ { 1 }\) has equation \(x + y + z = 3\) and the plane \(\Pi _ { 2 }\) has equation \(2 x + 3 y - z = 4\) The planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) intersect in the line \(L\).

1. Find a cartesian equation for the line \(L\). The plane \(\Pi _ { 3 }\) has equation $$\text { r. } \left( \begin{array} { r } 5 \\ - 4 \\ 4 \end{array} \right) = 12$$ The line \(L\) meets the plane \(\Pi _ { 3 }\) at the point \(A\).
2. Find the coordinates of \(A\).
3. Find the acute angle between \(\overrightarrow { O A }\) and the line \(L\), where \(O\) is the origin. Give your answer in degrees to one decimal place.

9 The lines \(l\) and \(m\) have equations \(\mathbf { r } = 3 \mathbf { i } - 2 \mathbf { j } + \mathbf { k } + \lambda ( - \mathbf { i } + 2 \mathbf { j } + \mathbf { k } )\) and \(\mathbf { r } = 4 \mathbf { i } + 4 \mathbf { j } + 2 \mathbf { k } + \mu ( a \mathbf { i } + b \mathbf { j } - \mathbf { k } )\) respectively, where \(a\) and \(b\) are constants.

1. Given that \(l\) and \(m\) intersect, show that $$2 a - b = 4 .$$
2. Given also that \(l\) and \(m\) are perpendicular, find the values of \(a\) and \(b\).
3. When \(a\) and \(b\) have these values, find the position vector of the point of intersection of \(l\) and \(m\).

9 Two lines \(l\) and \(m\) have equations \(\mathbf { r } = 3 \mathbf { i } + 2 \mathbf { j } + 5 \mathbf { k } + s ( 4 \mathbf { i } - \mathbf { j } + 3 \mathbf { k } )\) and \(\mathbf { r } = \mathbf { i } - \mathbf { j } - 2 \mathbf { k } + t ( - \mathbf { i } + 2 \mathbf { j } + 2 \mathbf { k } )\) respectively.

1. Show that \(l\) and \(m\) are perpendicular.
2. Show that \(l\) and \(m\) intersect and state the position vector of the point of intersection.
3. Show that the length of the perpendicular from the origin to the line \(m\) is \(\frac { 1 } { 3 } \sqrt { 5 }\).

6. $$\mathbf { A } = \left( \begin{array} { r r r } 1 & - 1 & 1 \\ 1 & 1 & 1 \\ 1 & 2 & a \end{array} \right) \quad a \neq 1$$

1. Find \(\mathbf { A } ^ { - 1 }\) in terms of \(a\).
   . The straight line \(l _ { 1 }\) is mapped onto the straight line \(l _ { 2 }\) by the transformation represented by the matrix \(\mathbf { B }\). $$\mathbf { B } = \left( \begin{array} { r r r } 1 & - 1 & 1 \\ 1 & 1 & 1 \\ 1 & 2 & 4 \end{array} \right)$$ The equation of \(l _ { 2 }\) is $$( \mathbf { r } - ( 12 \mathbf { i } + 4 \mathbf { j } + 6 \mathbf { k } ) ) \times ( - 6 \mathbf { i } + 2 \mathbf { j } + 3 \mathbf { k } ) = \mathbf { 0 }$$
2. Find a vector equation for the line \(l _ { 1 }\)

8. The plane \(\Pi _ { 1 }\) has equation $$x - 5 y + 3 z = 11$$ The plane \(\Pi _ { 2 }\) has equation $$3 x - 2 y + 2 z = 7$$ The planes \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) intersect in the line \(l\).

1. Find a vector equation for \(l\), giving your answer in the form \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b }\) where \(\mathbf { a }\) and \(\mathbf { b }\) are constant vectors and \(\lambda\) is a scalar parameter. The point \(P ( 2,0,3 )\) lies on \(\Pi _ { 1 }\) The line \(m\), which passes through \(P\), is parallel to \(l\). The point \(Q ( 3,2,1 )\) lies on \(\Pi _ { 2 }\) The line \(n\), which passes through \(Q\), is also parallel to \(l\).
2. Find, in exact simplified form, the shortest distance between \(m\) and \(n\).
   

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1. In this question you must show all stages of your working. Solutions relying entirely on calculator technology are not acceptable.

$$\mathbf { M } = \left( \begin{array} { r r r } 2 & 0 & 3 \\ 0 & - 4 & - 3 \\ 0 & - 4 & 0 \end{array} \right)$$ Given that \(\mathbf { M }\) has exactly two distinct eigenvalues \(\lambda _ { 1 }\) and \(\lambda _ { 2 }\) where \(\lambda _ { 1 } < \lambda _ { 2 }\)

1. determine a normalised eigenvector corresponding to the eigenvalue \(\lambda _ { 1 }\) The line \(l _ { 1 }\) has equation \(\mathbf { r } = \left( \begin{array} { r } 4 \\ - 1 \\ 0 \end{array} \right) + \mu \left( \begin{array} { r } 2 \\ 0 \\ - 1 \end{array} \right)\), where \(\mu\) is a scalar parameter.
   The transformation \(T\) is represented by \(\mathbf { M }\).
   The line \(l _ { 1 }\) is transformed by \(T\) to the line \(l _ { 2 }\)
2. Determine a vector equation for \(l _ { 2 }\), giving your answer in the form \(\mathbf { r } \times \mathbf { b } = \mathbf { c }\) where \(\mathbf { b }\) and \(\mathbf { c }\) are constant vectors.

6. \(\mathbf { M } = \left( \begin{array} { c c c } 1 & 0 & 3 \\ 0 & - 2 & 1 \\ k & 0 & 1 \end{array} \right)\), where \(k\) is a constant. Given that \(\left( \begin{array} { l } 6 \\ 1 \\ 6 \end{array} \right)\) is an eigenvector of \(\mathbf { M }\),

1. find the eigenvalue of \(\mathbf { M }\) corresponding to \(\left( \begin{array} { l } 6 \\ 1 \\ 6 \end{array} \right)\),
2. show that \(k = 3\),
3. show that \(\mathbf { M }\) has exactly two eigenvalues. A transformation \(T : \mathbb { R } ^ { 3 } \rightarrow \mathbb { R } ^ { 3 }\) is represented by \(\mathbf { M }\).
   The transformation \(T\) maps the line \(l _ { 1 }\), with cartesian equations \(\frac { x - 2 } { 1 } = \frac { y } { - 3 } = \frac { z + 1 } { 4 }\), onto the line \(l _ { 2 }\).
4. Taking \(k = 3\), find cartesian equations of \(l _ { 2 }\).

1. The matrix \(\mathbf { M }\) is given by

$$\mathbf { M } = \left( \begin{array} { r r r } 2 & 1 & 0 \\ 1 & 2 & 0 \\ - 1 & 0 & 4 \end{array} \right)$$

1. Show that 4 is an eigenvalue of \(\mathbf { M }\), and find the other two eigenvalues.
2. For the eigenvalue 4, find a corresponding eigenvector. The straight line \(l _ { 1 }\) is mapped onto the straight line \(l _ { 2 }\) by the transformation represented by the matrix \(\mathbf { M }\). The equation of \(l _ { 1 }\) is \(( \mathbf { r } - \mathbf { a } ) \times \mathbf { b } = 0\), where \(\mathbf { a } = 3 \mathbf { i } + 2 \mathbf { j } - 2 \mathbf { k }\) and \(\mathbf { b } = \mathbf { i } - \mathbf { j } + 2 \mathbf { k }\).
3. Find a vector equation for the line \(l _ { 2 }\).

1. The plane \(\Pi _ { 1 }\) has vector equation

$$\mathbf { r } . ( 3 \mathbf { i } - 4 \mathbf { j } + 2 \mathbf { k } ) = 5$$

1. Find the perpendicular distance from the point \(( 6,2,12 )\) to the plane \(\Pi _ { 1 }\) The plane \(\Pi _ { 2 }\) has vector equation $$\mathbf { r } = \lambda ( 2 \mathbf { i } + \mathbf { j } + 5 \mathbf { k } ) + \mu ( \mathbf { i } - \mathbf { j } - 2 \mathbf { k } ) , \text { where } \lambda \text { and } \mu \text { are scalar parameters. }$$
2. Find the acute angle between \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\) giving your answer to the nearest degree.
3. Find an equation of the line of intersection of the two planes in the form \(\mathbf { r } \times \mathbf { a } = \mathbf { b }\), where \(\mathbf { a }\) and \(\mathbf { b }\) are constant vectors.

8. The plane \(\Pi _ { 1 }\) has vector equation \(\mathbf { r }\). \(\left( \begin{array} { l } 2 \\ 1 \\ 3 \end{array} \right) = 5\) The plane \(\Pi _ { 2 }\) has vector equation \(\mathbf { r } . \left( \begin{array} { r } - 1 \\ 2 \\ 4 \end{array} \right) = 7\)

1. Find a vector equation for the line of intersection of \(\Pi _ { 1 }\) and \(\Pi _ { 2 }\), giving your answer in the form \(\mathbf { r } = \mathbf { a } + \lambda \mathbf { b }\) where \(\mathbf { a }\) and \(\mathbf { b }\) are constant vectors and \(\lambda\) is a scalar parameter. The plane \(\Pi _ { 3 }\) has cartesian equation $$x - y + 2 z = 31$$
2. Using your answer to part (a), or otherwise, find the coordinates of the point of intersection of the planes \(\Pi _ { 1 } , \Pi _ { 2 }\) and \(\Pi _ { 3 }\) \includegraphics[max width=\textwidth, alt={}, center]{393fd7be-c8f5-4b83-a5c7-2de04987a039-16_104_77_2469_1804}

5. The points \(A , B\) and \(C\) have position vectors \(\left( \begin{array} { l } 1 \\ 3 \\ 2 \end{array} \right) , \left( \begin{array} { r } - 1 \\ 0 \\ 1 \end{array} \right)\) and \(\left( \begin{array} { l } 2 \\ 1 \\ 0 \end{array} \right)\) respectively.

1. Find a vector equation of the straight line \(A B\).
2. Find a cartesian form of the equation of the straight line \(A B\). The plane \(\Pi\) contains the points \(A , B\) and \(C\).
3. Find a vector equation of \(\Pi\) in the form r.n \(= p\).
4. Find the perpendicular distance from the origin to \(\Pi\).