4.04c Scalar product: calculate and use for angles

7. The line \(l _ { 1 }\) passes through the points \(A\) and \(B\) with position vectors ( \(3 \mathbf { i } + 6 \mathbf { j } - 8 \mathbf { k }\) ) and ( \(8 \mathbf { j } - 6 \mathbf { k }\) ) respectively, relative to a fixed origin.

1. Find a vector equation for \(l _ { 1 }\). The line \(l _ { 2 }\) has vector equation $$\mathbf { r } = ( - 2 \mathbf { i } + 10 \mathbf { j } + 6 \mathbf { k } ) + \mu ( 7 \mathbf { i } - 4 \mathbf { j } + 6 \mathbf { k } ) ,$$ where \(\mu\) is a scalar parameter.
2. Show that lines \(l _ { 1 }\) and \(l _ { 2 }\) intersect.
3. Find the coordinates of the point where \(l _ { 1 }\) and \(l _ { 2 }\) intersect. The point \(C\) lies on \(l _ { 2 }\) and is such that \(A C\) is perpendicular to \(A B\).
4. Find the position vector of \(C\).

7. Relative to a fixed origin, two lines have the equations
and $$\begin{aligned} & \mathbf { r } = \left( \begin{array} { c } 7 \\ 0 \\ - 3 \end{array} \right) + s \left( \begin{array} { c } 5 \\ 4 \\ - 2 \end{array} \right) \\ & \mathbf { r } = \left( \begin{array} { l } a \\ 6 \\ 3 \end{array} \right) + t \left( \begin{array} { c } - 5 \\ 14 \\ 2 \end{array} \right) , \end{aligned}$$ where \(a\) is a constant and \(s\) and \(t\) are scalar parameters.
Given that the two lines intersect,

1. find the position vector of their point of intersection,
2. find the value of \(a\). Given also that \(\theta\) is the acute angle between the lines,
3. find the value of \(\cos \theta\) in the form \(k \sqrt { 5 }\) where \(k\) is rational.

4. The line \(l _ { 1 }\) passes through the points \(P\) and \(Q\) with position vectors ( \(- \mathbf { i } - 8 \mathbf { j } + 3 \mathbf { k }\) ) and ( \(2 \mathbf { i } - 9 \mathbf { j } + \mathbf { k }\) ) respectively, relative to a fixed origin.

1. Find a vector equation for \(l _ { 1 }\). The line \(l _ { 2 }\) has the equation $$\mathbf { r } = ( 6 \mathbf { i } + a \mathbf { j } + b \mathbf { k } ) + t ( \mathbf { i } + 4 \mathbf { j } - \mathbf { k } )$$ and also passes through the point \(Q\).
2. Find the values of the constants \(a\) and \(b\).
3. Find, in degrees to 1 decimal place, the acute angle between lines \(l _ { 1 }\) and \(l _ { 2 }\).

7. Relative to a fixed origin, two lines have the equations
and $$\begin{aligned} & \mathbf { r } = \left( \begin{array} { c } 4 \\ 1 \\ 1 \end{array} \right) + s \left( \begin{array} { l } 1 \\ 4 \\ 5 \end{array} \right) \\ & \mathbf { r } = \left( \begin{array} { c } - 3 \\ 1 \\ - 6 \end{array} \right) + t \left( \begin{array} { l } 3 \\ a \\ b \end{array} \right) , \end{aligned}$$ where \(a\) and \(b\) are constants and \(s\) and \(t\) are scalar parameters.
Given that the two lines are perpendicular,

1. find a linear relationship between \(a\) and \(b\). Given also that the two lines intersect,
2. find the values of \(a\) and \(b\),
3. find the coordinates of the point where they intersect.

3 Verify that the vector \(\mathbf { d } - \mathbf { j } + 4 \mathbf { k }\) is perpendicular to the plane through the points \(\mathrm { A } ( 2,0,1 ) , \mathrm { B } ( 1,2,2 )\) and \(\mathrm { C } ( 0 , - 4,1 )\). Hence find the cartesian equation of the plane. [5]

5 The points A , B and C have coordinates \(( 2,0 , - 1 ) , ( 4,3 , - 6 )\) and \(( 9,3 , - 4 )\) respectively.

1. Show that AB is perpendicular to BC .
2. Find the area of triangle ABC .

6

1. Verify that the lines \(\left. \mathbf { r } = \begin{array} { r } - 5 \\ 3 \\ 4 \end{array} \right) + \lambda \left( \begin{array} { r } 3 \\ 0 \\ - 1 \end{array} \right)\) and \(\left. \left. \mathbf { r } = \begin{array} { r } - 1 \\ 4 \\ 2 \end{array} \right) + \mu - \begin{array} { r } 2 \\ - 1 \\ 0 \end{array} \right)\) meet at the point ( \(1,3,2\) ).
2. Find the acute angle between the lines.

1 A glass ornament OABCDEFG is a truncated pyramid on a rectangular base (see Fig. 7). All dimensions are in centimetres. \begin{figure}[h] \end{figure}

1. Write down the vectors \(\overrightarrow { \mathrm { CD } }\) and \(\overrightarrow { \mathrm { CB } }\).
2. Find the length of the edge CD.
3. Show that the vector \(4 \mathbf { i } + \mathbf { k }\) is perpendicular to the vectors \(\overrightarrow { \mathrm { CD } }\) and \(\overrightarrow { \mathrm { CB } }\). Hence find the cartesian equation of the plane BCDE.
4. Write down vector equations for the lines OG and AF . Show that they meet at the point P with coordinates (5, 10, 40). You may assume that the lines CD and BE also meet at the point P .
   The volume of a pyramid is \(\frac { 1 } { 3 } \times\) area of base × height.
5. Find the volumes of the pyramids POABC and PDEFG . Hence find the volume of the ornament.

3 Fig. 6 shows a lean-to greenhouse ABCDHEFG . With respect to coordinate axes Oxyz , the coordinates of the vertices are as shown. All distances are in metres. Ground level is the plane \(z = 0\). \begin{figure}[h] \end{figure}

1. Verify that the equation of the plane through \(\mathrm { A } , \mathrm { B }\) and E is \(x + 6 y + 12 = 0\). Hence, given that F lies in this plane, show that \(a = - 2 \frac { 1 } { 3 }\).
2. (A) Show that the vector \(\left( \begin{array} { r } 1 \\ - 6 \\ 0 \end{array} \right)\) is normal to the plane DHC.
   (B) Hence find the cartesian equation of this plane.
   (C) Given that G lies in the plane DHC , find \(b\) and the length FG .
3. Find the angle EFB . A straight wire joins point H to a point P which is half way between E and \(\mathrm { F } . \mathrm { Q }\) is a point two-thirds of the way along this wire, so that \(\mathrm { HQ } = 2 \mathrm { QP }\).
4. Find the height of Q above the ground.

4 A computer-controlled machine can be programmed to make ats by entering the equation of the plane of the cut, and to drill holes by entering the equation of the line of the hole. A \(20 \mathrm {~cm} \times 30 \mathrm {~cm} \times 30 \mathrm {~cm}\) cuboid is to be at and drilled. The cuboid is positioned relative to \(x\)-, \(y ^ { 2 }\) and z-axes as shown in Fig. 8.1. \begin{figure}[h] \end{figure} \begin{figure}[h] \end{figure} Fkst, a plane out is made to remove the comer at \(E\). The cut goes through the points \(P . Q\) and \(R\), which are the midpoints of the sides \(\mathrm { ED } , \mathrm { EA }\) and EF respectively.

1. Write down the coordinates of \(\mathbf { P } , \mathbf { Q }\) and \(\mathbf { R }\). $$\text { Henceshowlhat } \mathbb { N } ^ { 1 } \left\{ \begin{array} { l } 1 \\ 0 \\ 0 \end{array} \right\} \text { and } \left\{ \begin{array} { l } 1 \\ 0 \end{array} \right\}$$ (;) Show that the veeto, \(\binom { \text { I } } { \text { ) } }\) is pc,pcndicula, to the plone through \(\mathrm { P } , \mathrm { Q }\), nd R
   Hence find the Cartesian equation of this plane. A hole is then drilled perpendicular to lriangle \(P Q R\), as shown in Fig. 82. The hole passes through the triangle at the point \(T\) which divides the line \(P S\) in the ratio 2 : \(I\), where \(S\) is the midpoint of \(Q R\).
2. Write down the coordinates of S , and show that the point T has coordinates \(( - 5,16,25 )\).
3. Write down a vector equation of the line of the drill hole. Hence determine whether or not this line passes through C .

1 Fig. 6 shows a lean-to greenhouse ABCDHEFG . With respect to coordinate axes \(\mathrm { O } x y z\), the coordinates of the vertices are as shown. All distances are in metres. Ground level is the plane \(z = 0\). \begin{figure}[h] \end{figure}

1. Verify that the equation of the plane through \(\mathrm { A } , \mathrm { B }\) and E is \(x + 6 y + 12 = 0\). Hence, given that F lies in this plane, show that \(a = - 2 \frac { 1 } { 3 }\).
2. (A) Show that the vector \(\left( \begin{array} { r } 1 \\ - 6 \\ 0 \end{array} \right)\) is normal to the plane DHC.
   (B) Hence find the cartesian equation of this plane.
   (C) Given that G lies in the plane DHC , find \(b\) and the length FG .
3. Find the angle EFB . A straight wire joins point H to a point P which is half way between E and \(\mathrm { F } . \mathrm { Q }\) is a point two-thirds of the way along this wire, so that \(\mathrm { HQ } = 2 \mathrm { QP }\).
4. Find the height of Q above the ground.

2 Fig. 7 shows a tetrahedron ABCD . The coordinates of the vertices, with respect to axes Oxyz , are \(\mathrm { A } ( - 3,0,0 ) , \mathrm { B } ( 2,0 , - 2 ) , \mathrm { C } ( 0,4,0 )\) and \(\mathrm { D } ( 0,4,5 )\). \begin{figure}[h] \end{figure}

1. Find the length of the edges AB and AC , and the size of the angle CAB . Hence calculate the area of triangle ABC .
2. (A) Verify that \(4 \mathbf { i } - 3 \mathbf { j } + 10 \mathbf { k }\) is normal to the plane ABC .
   (B) Hence find the equation of this plane.
3. Write down a vector equation for the line through D perpendicular to the plane ABC . Hence find the point of intersection of this line with the plane ABC . The volume of a tetrahedron is \(\frac { 1 } { 3 } \times\) area of base × height.
4. Find the volume of the tetrahedron ABCD .
5. Find a vector equation of the line \(l\) joining the points \(( 0,1,3 )\) and \(( - 2,2,5 )\).
6. Find the point of intersection of the line \(l\) with the plane \(x + 3 y + 2 z = 4\).
7. Find the acute angle between the line \(l\) and the normal to the plane.

1 With respect to cartesian coordinates Oxyz, a laser beam ABC is fired from the point \(\mathrm { A } ( 1,2,4 )\), and is reflected at point B off the plane with equation \(x + 2 y - 3 z = 0\), as shown in Fig. 8. \(\mathrm { A } ^ { \prime }\) is the point \(( 2,4,1 )\), and \(M\) is the midpoint of \(\mathrm { AA } ^ { \prime }\). \begin{figure}[h] \end{figure}

1. Show that \(\mathrm { AA } ^ { \prime }\) is perpendicular to the plane \(x + 2 y - 3 z = 0\), and that M lies in the plane. The vector equation of the line AB is \(\mathbf { r } = \left( \begin{array} { l } 1 \\ 2 \\ 4 \end{array} \right) + \lambda \left( \begin{array} { r } 1 \\ - 1 \\ 2 \end{array} \right)\).
2. Find the coordinates of B , and a vector equation of the line \(\mathrm { A } ^ { \prime } \mathrm { B }\).
3. Given that \(\mathrm { A } ^ { \prime } \mathrm { BC }\) is a straight line, find the angle \(\theta\).
4. Find the coordinates of the point where BC crosses the Oxz plane (the plane containing the \(x\) - and \(z\)-axes)

2 A piece of cloth ABDC is attached to the tops of vertical poles \(\mathrm { AE } , \mathrm { BF } , \mathrm { DG }\) and CH , where \(\mathrm { E } , \mathrm { F } , \mathrm { G }\) and H are at ground level (see Fig. 7). Coordinates are as shown, with lengths in metres. The length of pole DG is \(k\) metres. \begin{figure}[h] \end{figure}

1. Write down the vectors \(\overrightarrow { \mathrm { AB } }\) and \(\overrightarrow { \mathrm { AC } }\). Hence calculate the angle BAC .
2. Verify that the equation of the plane ABC is \(x + y - 2 z + d = 0\), where \(d\) is a constant to be determined. Calculate the acute angle the plane makes with the horizontal plane.
3. Given that \(\mathrm { A } , \mathrm { B } , \mathrm { D }\) and C are coplanar, show that \(k = 3\). Hence show that ABDC is a trapezium, and find the ratio of CD to AB .

3 A straight pipeline AB passes through a mountain. With respect to axes \(\mathrm { O } x y z\), with \(\mathrm { O } x\) due East, Oy due North and \(\mathrm { O } z\) vertically upwards, A has coordinates \(( - 200,100,0 )\) and B has coordinates \(( 100,200,100 )\), where units are metres.

1. Verify that \(\left. \overrightarrow { \mathrm { AB } } = \begin{array} { l } 300 \\ 100 \\ 100 \end{array} \right)\) and find the length of the pipeline.
   [0pt] [3]
2. Write down a vector equation of the line AB , and calculate the angle it makes with the vertical.
   [0pt] [6]
   A thin flat layer of hard rock runs through the mountain. The equation of the plane containing this layer is \(x + 2 y + 3 z = 320\).
3. Find the coordinates of the point where the pipeline meets the layer of rock.
4. By calculating the angle between the line AB and the normal to the plane of the layer, find the angle at which the pipeline cuts through the layer.

4 When a light ray passes from air to glass, it is deflected through an angle. The light ray ABC starts at point \(\mathrm { A } ( 1,2,2 )\), and enters a glass object at point \(\mathrm { B } ( 0,0,2 )\). The surface of the glass object is a plane with normal vector \(\mathbf { n }\). Fig. 7 shows a cross-section of the glass object in the plane of the light ray and \(\mathbf { n }\). \begin{figure}[h] \end{figure}

1. Find the vector \(\overrightarrow { \mathrm { AB } }\) and a vector equation of the line AB . The surface of the glass object is a plane with equation \(x + z = 2\). AB makes an acute angle \(\theta\) with the normal to this plane.
2. Write down the normal vector \(\mathbf { n }\), and hence calculate \(\theta\), giving your answer in degrees. The line BC has vector equation \(\mathbf { r } = \left( \begin{array} { l } 0 \\ 0 \\ 2 \end{array} \right) + \mu \left( \begin{array} { l } - 2 \\ - 2 \\ - 1 \end{array} \right)\). This line makes an acute angle \(\phi\) with the
   normal to the plane. normal to the plane.
3. Show that \(\phi = 45 ^ { \circ }\).
4. Snell's Law states that \(\sin \theta = k \sin \phi\), where \(k\) is a constant called the refractive index. Find \(k\). The light ray leaves the glass object through a plane with equation \(x + z = - 1\). Units are centimetres.
5. Find the point of intersection of the line BC with the plane \(x + z = - 1\). Hence find the distance the light ray travels through the glass object.

3 A triangle ABC has vertices \(\mathrm { A } ( - 2,4,1 ) , \mathrm { B } ( 2,3,4 )\) and \(\mathrm { C } ( 4,8,3 )\). By calculating a suitable scalar product, show that angle ABC is a right angle. Hence calculate the area of the triangle. [6]

5.The lines \(L _ { 1 }\) and \(L _ { 2 }\) have vector equations \(L _ { 1 } : \quad \mathbf { r } = - 2 \mathbf { i } + 11.5 \mathbf { j } + \lambda ( 3 \mathbf { i } - 4 \mathbf { j } - \mathbf { k } )\), \(L _ { 2 } : \quad \mathbf { r } = 11.5 \mathbf { i } + 3 \mathbf { j } + 8.5 \mathbf { k } + \mu ( 7 \mathbf { i } + 8 \mathbf { j } - 11 \mathbf { k } )\),
where \(\lambda\) and \(\mu\) are parameters.

1. Show that \(L _ { 1 }\) and \(L _ { 2 }\) do not intersect.
2. Show that the vector \(( 2 \mathbf { i } + \mathbf { j } + 2 \mathbf { k } )\) is perpendicular to both \(L _ { 1 }\) and \(L _ { 2 }\) . The point \(A\) lies on \(L _ { 1 }\) ,the point \(B\) lies on \(L _ { 2 }\) and \(A B\) is perpendicular to both \(L _ { 1 }\) and \(L _ { 2 }\) .
3. Find the position vector of the point \(A\) and the position vector of the point \(B\) .
   (8) \includegraphics[max width=\textwidth, alt={}, center]{0df09d8a-7478-4679-b117-128ee226db6a-4_554_1017_404_571} Figure 1 shows a sketch of part of the curve \(C\) with equation $$y = \sin ( \ln x ) , \quad x \geq 1 .$$ The point \(Q\) ,on \(C\) ,is a maximum.

4. \begin{figure}[h] \end{figure} Figure 1 shows a cuboid \(O A B C D E F G\), where \(O\) is the origin, \(A\) has position vector \(5 \mathbf { i } , C\) has position vector \(10 \mathbf { j }\) and \(D\) has position vector \(20 \mathbf { k }\).

1. Find the cosine of angle \(C A F\). Given that the point \(X\) lies on \(A C\) and that \(F X\) is perpendicular to \(A C\),
2. find the position vector of point \(X\) and the distance \(F X\). The line \(l _ { 1 }\) passes through \(O\) and through the midpoint of the face \(A B F E\). The line \(l _ { 2 }\) passes through \(A\) and through the midpoint of the edge \(F G\).
3. Show that \(l _ { 1 }\) and \(l _ { 2 }\) intersect and find the coordinates of the point of intersection.

3.The lines \(L _ { 1 }\) and \(L _ { 2 }\) have equations given by \(L _ { 1 } : \quad \mathbf { r } = \left( \begin{array} { c } - 7 \\ 7 \\ 1 \end{array} \right) + \lambda \left( \begin{array} { c } 2 \\ 0 \\ - 3 \end{array} \right)\) and \(L _ { 2 } : \quad \mathbf { r } = \left( \begin{array} { c } 7 \\ p \\ - 6 \end{array} \right) + \mu \left( \begin{array} { c } 10 \\ - 4 \\ - 1 \end{array} \right)\) where \(\lambda\) and \(\mu\) are parameters and \(p\) is a constant.
The two lines intersect at the point \(C\) .

1. Find
   1. the value of \(p\) ,
   2. the position vector of \(C\) .
2. Show that the point \(B\) with position vector \(\left( \begin{array} { c } - 13 \\ 11 \\ - 4 \end{array} \right)\) lies on \(L _ { 2 }\) . The point \(A\) with position vector \(\left( \begin{array} { c } - 7 \\ 7 \\ 1 \end{array} \right)\) lies on \(L _ { 1 }\) .
3. Find \(\cos ( \angle A C B )\) ,giving your answer as an exact fraction. The line \(L _ { 3 }\) bisects the angle \(A C B\) .
4. Find a vector equation of \(L _ { 3 }\) .

8 \includegraphics[max width=\textwidth, alt={}, center]{c1eee696-3d7f-410a-91a8-fa902309c117-14_485_716_262_676} In the diagram, \(O A B C\) is a pyramid in which \(O A = 2\) units, \(O B = 4\) units and \(O C = 2\) units. The edge \(O C\) is vertical, the base \(O A B\) is horizontal and angle \(A O B = 90 ^ { \circ }\). Unit vectors \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\) are parallel to \(O A\), \(O B\) and \(O C\) respectively. The midpoints of \(A B\) and \(B C\) are \(M\) and \(N\) respectively.

1. Express the vectors \(\overrightarrow { O N }\) and \(\overrightarrow { C M }\) in terms of \(\mathbf { i } , \mathbf { j }\) and \(\mathbf { k }\).
2. Calculate the angle between the directions of \(\overrightarrow { O N }\) and \(\overrightarrow { C M }\).
3. Show that the length of the perpendicular from \(M\) to \(O N\) is \(\frac { 3 } { 5 } \sqrt { 5 }\).

1 The plane \(p\) has equation \(\mathbf { r } . ( \mathbf { i } - 3 \mathbf { j } + 4 \mathbf { k } ) = 4\) and the line \(l _ { 1 }\) has equation \(\mathbf { r } = 2 \mathbf { j } - \mathbf { k } + t ( 3 \mathbf { i } + \mathbf { j } + 2 \mathbf { k } )\). The line \(l _ { 2 }\) is parallel to \(p\) and perpendicular to \(l _ { 1 }\), and passes through the point with position vector \(\mathbf { i } + 4 \mathbf { j } + 2 \mathbf { k }\). Find the equation of \(l _ { 2 }\), giving your answer in the form \(\mathbf { r } = \mathbf { a } + t \mathbf { b }\).

7 With respect to the origin \(O\), the position vectors of the points \(U , V\) and \(W\) are \(\mathbf { u } , \mathbf { v }\) and \(\mathbf { w }\) respectively. The mid-points of the sides \(V W , W U\) and \(U V\) of the triangle \(U V W\) are \(M , N\) and \(P\) respectively.

1. Show that \(\overrightarrow { U M } = \frac { 1 } { 2 } ( \mathbf { v } + \mathbf { w } - 2 \mathbf { u } )\).
2. Verify that the point \(G\) with position vector \(\frac { 1 } { 3 } ( \mathbf { u } + \mathbf { v } + \mathbf { w } )\) lies on \(U M\), and deduce that the lines \(U M , V N\) and \(W P\) intersect at \(G\).
3. Write down, in the form \(\mathbf { r } = \mathbf { a } + t \mathbf { b }\), an equation of the line through \(G\) which is perpendicular to the plane \(U V W\). (It is not necessary to simplify the expression for \(\mathbf { b }\).)
4. It is now given that \(\mathbf { u } = \left( \begin{array} { l } 1 \\ 0 \\ 0 \end{array} \right) , \mathbf { v } = \left( \begin{array} { l } 0 \\ 1 \\ 0 \end{array} \right)\) and \(\mathbf { w } = \left( \begin{array} { l } 0 \\ 0 \\ 1 \end{array} \right)\). Find the perpendicular distance from \(O\) to the plane \(U V W\).

6 The line \(l _ { 1 }\) has equation \(\mathbf { r } = \left( \begin{array} { r } 3 \\ 0 \\ - 2 \end{array} \right) + s \left( \begin{array} { r } 2 \\ 3 \\ - 4 \end{array} \right)\). The line \(l _ { 2 }\) has equation \(\mathbf { r } = \left( \begin{array} { l } 5 \\ 3 \\ 2 \end{array} \right) + t \left( \begin{array} { r } 0 \\ 1 \\ - 2 \end{array} \right)\).

1. Find the acute angle between \(l _ { 1 }\) and \(l _ { 2 }\).
2. Show that \(l _ { 1 }\) and \(l _ { 2 }\) are skew.
3. One of the numbers in the equation of line \(l _ { 1 }\) is changed so that the equation becomes \(\mathbf { r } = \left( \begin{array} { l } 3 \\ 0 \\ a \end{array} \right) + s \left( \begin{array} { r } 2 \\ 3 \\ - 4 \end{array} \right)\). Given that \(l _ { 1 }\) and \(l _ { 2 }\) now intersect, find \(a\).

2

1. Find, in the form \(\mathbf { r } = \mathbf { a } + t \mathbf { b }\), an equation of the line \(l\) through the points ( \(4,2,7\) ) and ( \(5 , - 4 , - 1\) ).
2. Find the acute angle between the line \(l\) and a line in the direction of the vector \(\left( \begin{array} { l } 1 \\ 2 \\ 3 \end{array} \right)\).