4.04g Vector product: a x b perpendicular vector

6 Relative to an origin \(O\), the position vector of \(A\) is \(3 \mathbf { i } + 2 \mathbf { j } - \mathbf { k }\) and the position vector of \(B\) is \(7 \mathbf { i } - 3 \mathbf { j } + \mathbf { k }\).

1. Show that angle \(O A B\) is a right angle.
2. Find the area of triangle \(O A B\).

10 With respect to the origin \(O\), the points \(A , B\) and \(C\) have position vectors given by $$\overrightarrow { O A } = \left( \begin{array} { r } 3 \\ - 2 \\ 4 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } 2 \\ - 1 \\ 7 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { r } 1 \\ - 5 \\ - 3 \end{array} \right) .$$ The plane \(m\) is parallel to \(\overrightarrow { O C }\) and contains \(A\) and \(B\).

1. Find the equation of \(m\), giving your answer in the form \(a x + b y + c z = d\).
2. Find the length of the perpendicular from \(C\) to the line through \(A\) and \(B\).

10 With respect to the origin \(O\), the points \(A , B\) and \(C\) have position vectors given by $$\overrightarrow { O A } = \left( \begin{array} { r } 3 \\ - 2 \\ 4 \end{array} \right) , \quad \overrightarrow { O B } = \left( \begin{array} { r } 2 \\ - 1 \\ 7 \end{array} \right) \quad \text { and } \quad \overrightarrow { O C } = \left( \begin{array} { r } 1 \\ - 5 \\ - 3 \end{array} \right)$$ The plane \(m\) is parallel to \(\overrightarrow { O C }\) and contains \(A\) and \(B\).

1. Find the equation of \(m\), giving your answer in the form \(a x + b y + c z = d\).
2. Find the length of the perpendicular from \(C\) to the line through \(A\) and \(B\).

9
The diagram shows three points \(A , B\) and \(C\) whose position vectors with respect to the origin \(O\) are given by \(\overrightarrow { O A } = \left( \begin{array} { r } 2 \\ - 1 \\ 2 \end{array} \right) , \overrightarrow { O B } = \left( \begin{array} { l } 0 \\ 3 \\ 1 \end{array} \right)\) and \(\overrightarrow { O C } = \left( \begin{array} { l } 3 \\ 0 \\ 4 \end{array} \right)\). The point \(D\) lies on \(B C\), between \(B\) and \(C\), and is such that \(C D = 2 D B\).

1. Find the equation of the plane \(A B C\), giving your answer in the form \(a x + b y + c z = d\).
2. Find the position vector of \(D\).
3. Show that the length of the perpendicular from \(A\) to \(O D\) is \(\frac { 1 } { 3 } \sqrt { } ( 65 )\).

6 The points \(A , B , C\) have position vectors $$\mathbf { i } + \mathbf { j } , \quad - \mathbf { i } + 2 \mathbf { j } + 4 \mathbf { k } , \quad - 2 \mathbf { i } + \mathbf { j } + 3 \mathbf { k } ,$$ respectively, relative to the origin \(O\).

1. Find the equation of the plane \(A B C\), giving your answer in the form \(a x + b y + c z = d\).
2. Find the perpendicular distance from \(O\) to the plane \(A B C\).
3. Find a vector equation of the common perpendicular to the lines \(O C\) and \(A B\).

6 The lines \(l\) and \(m\) have vector equations $$l : \quad \mathbf { r } = 2 \mathbf { i } + \mathbf { j } - 3 \mathbf { k } + \lambda ( - \mathbf { i } + 2 \mathbf { k } ) \quad \text { and } \quad m : \quad \mathbf { r } = 2 \mathbf { i } + \mathbf { j } - 3 \mathbf { k } + \mu ( 2 \mathbf { i } - \mathbf { j } + 5 \mathbf { k } ) .$$ Lines \(l\) and \(m\) intersect at the point \(P\).

1. State the coordinates of \(P\).
2. Find the exact value of the cosine of the acute angle between \(l\) and \(m\).
3. The point \(A\) on line \(I\) has coordinates ( \(0,1,1\) ). The point \(B\) on line \(m\) has coordinates ( \(0,2 , - 8\) ). Find the exact area of triangle \(A P B\).

7. Relative to a fixed origin \(O\), the point \(A\) has position vector ( \(2 \mathbf { i } - \mathbf { j } + 5 \mathbf { k }\) ), the point \(B\) has position vector \(( 5 \mathbf { i } + 2 \mathbf { j } + 10 \mathbf { k } )\), and the point \(D\) has position vector \(( - \mathbf { i } + \mathbf { j } + 4 \mathbf { k } )\). 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 \(l\).
3. Show that the size of the angle \(B A D\) is \(109 ^ { \circ }\), to the nearest degree. The points \(A , B\) and \(D\), together with a point \(C\), are the vertices of the parallelogram \(A B C D\), where \(\overrightarrow { A B } = \overrightarrow { D C }\).
4. Find the position vector of \(C\).
5. Find the area of the parallelogram \(A B C D\), giving your answer to 3 significant figures.
6. Find the shortest distance from the point \(D\) to the line \(l\), giving your answer to 3 significant figures.

7. The point \(A\) with coordinates ( \(- 3,7,2\) ) lies on a line \(l _ { 1 }\) The point \(B\) also lies on the line \(l _ { 1 }\) Given that \(\quad \overrightarrow { A B } = \left( \begin{array} { r } 4 \\ - 6 \\ 2 \end{array} \right)\),

1. find the coordinates of point \(B\). The point \(P\) has coordinates ( \(9,1,8\) )
2. Find the cosine of the angle \(P A B\), giving your answer as a simplified surd.
3. Find the exact area of triangle \(P A B\), giving your answer in its simplest form. The line \(l _ { 2 }\) passes through the point \(P\) and is parallel to the line \(l _ { 1 }\)
4. Find a vector equation for the line \(l _ { 2 }\) The point \(Q\) lies on the line \(l _ { 2 }\) Given that the line segment \(A P\) is perpendicular to the line segment \(B Q\),
5. find the coordinates of the point \(Q\).

9 With respect to a fixed origin \(O\), the points \(A ( - 1,5,1 ) , B ( 1,0,3 ) , C ( 2 , - 1,2 )\) and \(D ( 3,6 , - 1 )\) are the vertices of a tetrahedron.

1. Find the volume of the tetrahedron \(A B C D\). The plane \(\Pi\) contains the points \(A , B\) and \(C\).
2. Find a cartesian equation of \(\Pi\). The point \(T\) lies on the plane \(\Pi\). The line \(D T\) is perpendicular to \(\Pi\).
3. Find the exact coordinates of the point \(T\).

1. The skew lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations

$$l _ { 1 } : \mathbf { r } = ( \mathbf { i } + 2 \mathbf { j } - 5 \mathbf { k } ) + \lambda ( 5 \mathbf { i } + \mathbf { j } )$$ and $$l _ { 2 } : \mathbf { r } = ( 2 \mathbf { i } - 4 \mathbf { j } + 4 \mathbf { k } ) + \mu ( 8 \mathbf { i } - 2 \mathbf { j } + 3 \mathbf { k } )$$ where \(\lambda\) and \(\mu\) are scalar parameters.

1. Determine a vector that is perpendicular to both \(l _ { 1 }\) and \(l _ { 2 }\)
2. Determine an equation of the plane parallel to \(l _ { 1 }\) that contains \(l _ { 2 }\)
   1. in the form \(\mathbf { r } = \mathbf { a } + s \mathbf { b } + t \mathbf { c }\)
   2. in the form r.n \(= p\)
3. Determine the shortest distance between \(l _ { 1 }\) and \(l _ { 2 }\) Give your answer in simplest form.

2. \begin{figure}[h] \end{figure} The points \(A , B\) and \(C\) have position vectors \(\mathbf { a } , \mathbf { b }\) and \(\mathbf { c }\) respectively, relative to a fixed origin \(O\), as shown in Figure 1. It is given that $$\mathbf { a } = \mathbf { i } + \mathbf { j } , \quad \mathbf { b } = 3 \mathbf { i } - \mathbf { j } + \mathbf { k } \quad \text { and } \quad \mathbf { c } = 2 \mathbf { i } + \mathbf { j } - \mathbf { k } .$$ Calculate

1. \(\mathbf { b } \times \mathbf { c }\),
2. a.(b \(\times \mathbf { c ) }\),
3. the area of triangle \(O B C\),
4. the volume of the tetrahedron \(O A B C\).

2. \begin{figure}[h] \end{figure} The points \(A , B\) and \(C\) have position vectors \(\mathbf { a } , \mathbf { b }\) and \(\mathbf { c }\) respectively, relative to a fixed origin \(O\), as shown in Figure 1. It is given that $$\mathbf { a } = \mathbf { i } + \mathbf { j } , \quad \mathbf { b } = 3 \mathbf { i } - \mathbf { j } + \mathbf { k } \quad \text { and } \quad \mathbf { c } = 2 \mathbf { i } + \mathbf { j } - \mathbf { k } .$$ Calculate

1. \(\mathbf { b } \times \mathbf { c }\),
2. a.(b \(\times \mathbf { c ) }\),
3. the area of triangle \(O B C\),
4. the volume of the tetrahedron \(O A B C\).

3. The position vectors of the points \(A , B\) and \(C\) relative to an origin \(O\) are \(\mathbf { i } - 2 \mathbf { j } - 2 \mathbf { k } , 7 \mathbf { i } - 3 \mathbf { k }\) and \(4 \mathbf { i } + 4 \mathbf { j }\) respectively. Find

1. \(\overrightarrow { A C } \times \overrightarrow { B C }\),
2. the area of triangle \(A B C\),
3. an equation of the plane \(A B C\) in the form \(\mathbf { r } . \mathbf { n } = p\)

2. Two skew lines \(l _ { 1 }\) and \(l _ { 2 }\) have equations $$\begin{aligned} & l _ { 1 } : \mathbf { r } = ( \mathbf { i } - \mathbf { j } + \mathbf { k } ) + \lambda ( 4 \mathbf { i } + 3 \mathbf { j } + 2 \mathbf { k } ) \\ & l _ { 2 } : \mathbf { r } = ( 3 \mathbf { i } + 7 \mathbf { j } + 2 \mathbf { k } ) + \mu ( - 4 \mathbf { i } + 6 \mathbf { j } + \mathbf { k } ) \end{aligned}$$ respectively, where \(\lambda\) and \(\mu\) are real parameters.

1. Find a vector in the direction of the common perpendicular to \(l _ { 1 }\) and \(l _ { 2 }\)
2. Find the shortest distance between these two lines.

1 Four points have coordinates \(\mathrm { A } ( - 2 , - 3,2 ) , \mathrm { B } ( - 3,1,5 ) , \mathrm { C } ( k , 5 , - 2 )\) and \(\mathrm { D } ( 0,9 , k )\).

1. Find the vector product \(\overrightarrow { \mathrm { AB } } \times \overrightarrow { \mathrm { CD } }\).
2. For the case when AB is parallel to CD ,
   (A) state the value of \(k\),
   (B) find the shortest distance between the parallel lines AB and CD ,
   (C) find, in the form \(a x + b y + c z + d = 0\), the equation of the plane containing AB and CD .
3. When AB is not parallel to CD , find the shortest distance between the lines AB and CD , in terms of \(k\).
4. Find the value of \(k\) for which the line AB intersects the line CD , and find the coordinates of the point of intersection in this case.

1 A tetrahedron ABCD has vertices \(\mathrm { A } ( - 3,5,2 ) , \mathrm { B } ( 3,13,7 ) , \mathrm { C } ( 7,0,3 )\) and \(\mathrm { D } ( 5,4,8 )\).

1. Find the vector product \(\overrightarrow { \mathrm { AB } } \times \overrightarrow { \mathrm { AC } }\), and hence find the equation of the plane ABC .
2. Find the shortest distance from \(D\) to the plane \(A B C\).
3. Find the shortest distance between the lines AB and CD .
4. Find the volume of the tetrahedron ABCD . The plane \(P\) with equation \(3 x - 2 z + 5 = 0\) contains the point B , and meets the lines AC and AD at E and F respectively.
5. Find \(\lambda\) and \(\mu\) such that \(\overrightarrow { \mathrm { AE } } = \lambda \overrightarrow { \mathrm { AC } }\) and \(\overrightarrow { \mathrm { AF } } = \mu \overrightarrow { \mathrm { AD } }\). Deduce that E is between A and C , and that F is between A and D.
6. Hence, or otherwise, show that \(P\) divides the tetrahedron ABCD into two parts having volumes in the ratio 4 to 17.

1 Four points have coordinates $$\mathrm { A } ( 3,8,27 ) , \quad \mathrm { B } ( 5,9,25 ) , \quad \mathrm { C } ( 8,0,1 ) \quad \text { and } \quad \mathrm { D } ( 11 , p , p ) ,$$ where \(p\) is a constant.

1. Find the perpendicular distance from C to the line AB .
2. Find \(\overrightarrow { \mathrm { AB } } \times \overrightarrow { \mathrm { CD } }\) in terms of \(p\), and show that the shortest distance between the lines AB and CD is $$\frac { 21 | p - 5 | } { \sqrt { 17 p ^ { 2 } - 2 p + 26 } }$$
3. Find, in terms of \(p\), the volume of the tetrahedron ABCD .
4. State the value of \(p\) for which the lines AB and CD intersect, and find the coordinates of the point of intersection in this case. Option 2: Multi-variable calculus

2 In this question, \(L\) is the straight line with equation \(\mathbf { r } = \left( \begin{array} { r } 2 \\ 1 \\ - 1 \end{array} \right) + \lambda \left( \begin{array} { r } - 2 \\ 2 \\ 1 \end{array} \right)\), and \(\mathrm { g } ( x , y , z ) = \left( x y + z ^ { 2 } \right) \mathrm { e } ^ { x - 2 y }\).

1. Find \(\frac { \partial \mathrm { g } } { \partial x } , \frac { \partial \mathrm {~g} } { \partial y }\) and \(\frac { \partial \mathrm { g } } { \partial z }\).
2. Show that the normal to the surface \(\mathrm { g } ( x , y , z ) = 3\) at the point \(( 2,1 , - 1 )\) is the line \(L\). On the line \(L\), there are two points at which \(\mathrm { g } ( x , y , z ) = 0\).
3. Show that one of these points is \(\mathrm { P } ( 0,3,0 )\), and find the coordinates of the other point Q .
4. Show that, if \(x = - 2 \mu , y = 3 + 2 \mu , z = \mu\), and \(\mu\) is small, then $$\mathrm { g } ( x , y , z ) \approx - 6 \mu \mathrm { e } ^ { - 6 }$$ You are given that \(h\) is a small number.
5. There is a point on \(L\), close to P , at which \(\mathrm { g } ( x , y , z ) = h\). Show that this point is approximately $$\left( \frac { 1 } { 3 } \mathrm { e } ^ { 6 } h , 3 - \frac { 1 } { 3 } \mathrm { e } ^ { 6 } h , - \frac { 1 } { 6 } \mathrm { e } ^ { 6 } h \right)$$
6. Find the approximate coordinates of the point on \(L\), close to Q , at which \(\mathrm { g } ( x , y , z ) = h\).

1 A mine contains several underground tunnels beneath a hillside. The hillside is a plane, all the tunnels are straight and the width of the tunnels may be neglected. A coordinate system is chosen with the \(z\)-axis pointing vertically upwards and the units are metres. Three points on the hillside have coordinates \(\mathrm { A } ( 15 , - 60,20 )\), \(B ( - 75,100,40 )\) and \(C ( 18,138,35.6 )\).

1. Find the vector product \(\overrightarrow { \mathrm { AB } } \times \overrightarrow { \mathrm { AC } }\) and hence show that the equation of the hillside is \(2 x - 2 y + 25 z = 650\). The tunnel \(T _ { \mathrm { A } }\) begins at A and goes in the direction of the vector \(15 \mathbf { i } + 14 \mathbf { j } - 2 \mathbf { k }\); the tunnel \(T _ { \mathrm { C } }\) begins at C and goes in the direction of the vector \(8 \mathbf { i } + 7 \mathbf { j } - 2 \mathbf { k }\). Both these tunnels extend a long way into the ground.
2. Find the least possible length of a tunnel which connects B to a point in \(T _ { \mathrm { A } }\).
3. Find the least possible length of a tunnel which connects a point in \(T _ { \mathrm { A } }\) to a point in \(T _ { \mathrm { C } }\).
4. A tunnel starts at B , passes through the point ( \(18,138 , p\) ) vertically below C , and intersects \(T _ { \mathrm { A } }\) at the point Q . Find the value of \(p\) and the coordinates of Q .

1 Three points have coordinates \(\mathrm { A } ( - 3,12 , - 7 ) , \mathrm { B } ( - 2,6,9 ) , \mathrm { C } ( 6,0 , - 10 )\). The plane \(P\) passes through the points \(\mathrm { A } , \mathrm { B }\) and C .

1. Find the vector product \(\overrightarrow { \mathrm { AB } } \times \overrightarrow { \mathrm { AC } }\). Hence or otherwise find an equation for the plane \(P\) in the form \(a x + b y + c z = d\). The plane \(Q\) has equation \(6 x + 3 y + 2 z = 32\). The perpendicular from A to the plane \(Q\) meets \(Q\) at the point D. The planes \(P\) and \(Q\) intersect in the line \(L\).
2. Find the distance AD .
3. Find an equation for the line \(L\).
4. Find the shortest distance from A to the line \(L\).
5. Find the volume of the tetrahedron ABCD .

4. Relative to a fixed origin, \(O\), the points \(A\) and \(B\) have position vectors \(\left( \begin{array} { c } 1 \\ 5 \\ - 1 \end{array} \right)\) and \(\left( \begin{array} { c } 6 \\ 3 \\ - 6 \end{array} \right)\) respectively. Find, in exact, simplified form,

1. the cosine of \(\angle A O B\),
2. the area of triangle \(O A B\),
3. the shortest distance from \(A\) to the line \(O B\).

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.

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 \(U V W\).