SL Paper 2
Let \(u = 6i + 3j + 6k\) and \(v = 2i + 2j + k\).
Find
(i) \(u \bullet v\);
(ii) \(\left| {{u}} \right|\);
(iii) \(\left| {{v}} \right|\).
Find the angle between \({{u}}\) and \({{v}}\).
Two lines with equations \({{\boldsymbol{r}}_1} = \left( {\begin{array}{*{20}{c}}
2\\
3\\
{ - 1}
\end{array}} \right) + s\left( {\begin{array}{*{20}{c}}
5\\
{ - 3}\\
2
\end{array}} \right)\) and \({{\boldsymbol{r}}_2} = \left( {\begin{array}{*{20}{c}}
9\\
2\\
2
\end{array}} \right) + t\left( {\begin{array}{*{20}{c}}
{ - 3}\\
5\\
{ - 1}
\end{array}} \right)\) intersect at the point P. Find the coordinates of P.
Let \({\boldsymbol{v}} = \left( {\begin{array}{*{20}{c}}
2\\
{ - 3}\\
6
\end{array}} \right)\) and \({\boldsymbol{w}} = \left( {\begin{array}{*{20}{c}}
k\\
{ - 2}\\
4
\end{array}} \right)\) , for \(k > 0\) . The angle between v and w is \(\frac{\pi }{3}\) .
Find the value of \(k\) .
Let \(\overrightarrow {{\text{AB}}} = \left( {\begin{array}{*{20}{c}} 4 \\ 1 \\ 2 \end{array}} \right)\).
Find \(\left| {\overrightarrow {{\text{AB}}} } \right|\).
Let \(\overrightarrow {{\text{AC}}} = \left( {\begin{array}{*{20}{c}} 3 \\ 0 \\ 0 \end{array}} \right)\). Find \({\rm{B\hat AC}}\).
In this question, distance is in metres.
Toy airplanes fly in a straight line at a constant speed. Airplane 1 passes through a point A.
Its position, p seconds after it has passed through A, is given by \(\left( {\begin{array}{*{20}{c}}
x\\
y\\
z
\end{array}} \right) = \left( {\begin{array}{*{20}{c}}
3\\
{ - 4}\\
0
\end{array}} \right) + p\left( {\begin{array}{*{20}{c}}
{ - 2}\\
3\\
1
\end{array}} \right)\) .
(i) Write down the coordinates of A.
(ii) Find the speed of the airplane in \({\text{m}}{{\text{s}}^{ - 1}}\).
After seven seconds the airplane passes through a point B.
(i) Find the coordinates of B.
(ii) Find the distance the airplane has travelled during the seven seconds.
Airplane 2 passes through a point C. Its position q seconds after it passes through C is given by \(\left( {\begin{array}{*{20}{c}}
x\\
y\\
z
\end{array}} \right) = \left( {\begin{array}{*{20}{c}}
2\\
{ - 5}\\
8
\end{array}} \right) + q\left( {\begin{array}{*{20}{c}}
{ - 1}\\
2\\
a
\end{array}} \right),a \in \mathbb{R}\) .
The angle between the flight paths of Airplane 1 and Airplane 2 is \({40^ \circ }\) . Find the two values of a.
Consider the points P(2, −1, 5) and Q(3, − 3, 8). Let \({L_1}\) be the line through P and Q.
Show that \(\overrightarrow {{\rm{PQ}}} = \left( {\begin{array}{*{20}{c}}
1\\
{ - 2}\\
3
\end{array}} \right)\) .
The line \({L_1}\) may be represented by \({\boldsymbol{r}} = \left( {\begin{array}{*{20}{c}}
3 \\
{ - 3} \\
8
\end{array}} \right) + s\left( {\begin{array}{*{20}{c}}
1 \\
{ - 2} \\
3
\end{array}} \right)\) .
(i) What information does the vector \(\left( {\begin{array}{*{20}{c}}
3\\
{ - 3}\\
8
\end{array}} \right)\) give about \({L_1}\) ?
(ii) Write down another vector representation for \({L_1}\) using \(\left( {\begin{array}{*{20}{c}}
3\\
{ - 3}\\
8
\end{array}} \right)\) .
The point \({\text{T}}( - 1{\text{, }}5{\text{, }}p)\) lies on \({L_1}\) .
Find the value of \(p\) .
The point T also lies on \({L_2}\) with equation \(\left( {\begin{array}{*{20}{c}}
x\\
y\\
z
\end{array}} \right) = \left( {\begin{array}{*{20}{c}}
{ - 3}\\
9\\
2
\end{array}} \right) + t\left( {\begin{array}{*{20}{c}}
1\\
{ - 2}\\
q
\end{array}} \right)\) .
Show that \(q = - 3\) .
Let \(\theta \) be the obtuse angle between \({L_1}\) and \({L_2}\) . Calculate the size of \(\theta \) .
Consider the points \({\text{A }}(1,{\text{ }}5,{\text{ }} - 7)\) and \({\text{B }}( - 9,{\text{ }}9,{\text{ }} - 6)\).
Let C be a point such that \(\overrightarrow {{\text{AC}}} = \left( {\begin{array}{*{20}{c}} 6 \\ { - 4} \\ 0 \end{array}} \right)\).
The line \(L\) passes through B and is parallel to (AC).
Find \(\overrightarrow {{\text{AB}}} \).
Find the coordinates of C.
Write down a vector equation for \(L\).
Given that \(\left| {\overrightarrow {{\text{AB}}} } \right| = k\left| {\overrightarrow {{\text{AC}}} } \right|\), find \(k\).
The point D lies on \(L\) such that \(\left| {\overrightarrow {{\text{AB}}} } \right| = \left| {\overrightarrow {{\text{BD}}} } \right|\). Find the possible coordinates of D.
The point O has coordinates (0 , 0 , 0) , point A has coordinates (1 , – 2 , 3) and point B has coordinates (– 3 , 4 , 2) .
(i) Show that \(\overrightarrow {{\rm{AB}}} = \left( {\begin{array}{*{20}{c}}
{ - 4}\\
6\\
{ - 1}
\end{array}} \right)\) .
(ii) Find \({\rm{B}}\widehat {\rm{A}}{\rm{O}}\) .
The line \({L_1}\) has equation \(\left( {\begin{array}{*{20}{c}}
x\\
y\\
z
\end{array}} \right) = \left( {\begin{array}{*{20}{c}}
{ - 3}\\
4\\
2
\end{array}} \right) + s\left( {\begin{array}{*{20}{c}}
{ - 4}\\
6\\
{ - 1}
\end{array}} \right)\) .
Write down the coordinates of two points on \({L_1}\) .
The line \({L_2}\) passes through A and is parallel to \(\overrightarrow {{\rm{OB}}} \) .
(i) Find a vector equation for \({L_2}\) , giving your answer in the form \({\boldsymbol{r}} = {\boldsymbol{a}} + t{\boldsymbol{b}}\) .
(ii) Point \(C(k, - k,5)\) is on \({L_2}\) . Find the coordinates of C.
The line \({L_3}\) has equation \(\left( {\begin{array}{*{20}{c}}
x\\
y\\
z
\end{array}} \right) = \left( {\begin{array}{*{20}{c}}
3\\
{ - 8}\\
0
\end{array}} \right) + p\left( {\begin{array}{*{20}{c}}
1\\
{ - 2}\\
{ - 1}
\end{array}} \right)\) and passes through the point C.
Find the value of p at C.
Consider the lines \({L_1}\) and \({L_2}\) with equations \({L_1}\) : \(\boldsymbol{r}=\left( \begin{array}{c}11\\8\\2\end{array} \right) + s\left( \begin{array}{c}4\\3\\ - 1\end{array} \right)\) and \({L_2}\) : \(\boldsymbol{r} = \left( \begin{array}{c}1\\1\\ - 7\end{array} \right) + t\left( \begin{array}{c}2\\1\\11\end{array} \right)\).
The lines intersect at point \(\rm{P}\).
Find the coordinates of \({\text{P}}\).
Show that the lines are perpendicular.
The point \({\text{Q}}(7, 5, 3)\) lies on \({L_1}\). The point \({\text{R}}\) is the reflection of \({\text{Q}}\) in the line \({L_2}\).
Find the coordinates of \({\text{R}}\).
The following diagram shows the cuboid (rectangular solid) OABCDEFG, where O is the origin, and \(\overrightarrow {{\rm{OA}}} = 4\boldsymbol{i}\) , \(\overrightarrow {{\rm{OC}}} = 3\boldsymbol{j}\) , \(\overrightarrow {{\rm{OD}}} = 2\boldsymbol{k}\) .
(i) Find \(\overrightarrow {{\rm{OB}}} \) .
(ii) Find \(\overrightarrow {{\rm{OF}}} \) .
(iii) Show that \(\overrightarrow {{\rm{AG}}} = - 4{\boldsymbol{i}} + 3{\boldsymbol{j}} + 2{\boldsymbol{k}}\) .
Write down a vector equation for
(i) the line OF;
(ii) the line AG.
Find the obtuse angle between the lines OF and AG.
Consider the points A(\(5\), \(2\), \(1\)) , B(\(6\), \(5\), \(3\)) , and C(\(7\), \(6\), \(a + 1\)) , \(a \in{\mathbb{R}}\) .
Let \({\rm{q}}\) be the angle between \(\overrightarrow {{\rm{AB}}} \) and \(\overrightarrow {{\rm{AC}}} \) .
Find
(i) \(\overrightarrow {{\rm{AB}}} \) ;
(ii) \(\overrightarrow {{\rm{AC}}} \) .
Find the value of \(a\) for which \({\rm{q}} = \frac{\pi }{2}\) .
i. Show that \(\cos q = \frac{{2a + 14}}{{\sqrt {14{a^2} + 280} }}\) .
ii. Hence, find the value of a for which \({\rm{q}} = 1.2\) .
Hence, find the value of a for which \({\rm{q}} = 1.2\) .
Let \({\boldsymbol{v}} = 3{\boldsymbol{i}} + 4{\boldsymbol{j}} + {\boldsymbol{k}}\) and \({\boldsymbol{w}} = {\boldsymbol{i}} + 2{\boldsymbol{j}} - 3{\boldsymbol{k}}\) . The vector \({\boldsymbol{v}} + p{\boldsymbol{w}}\) is perpendicular to w. Find the value of p.
The points A and B lie on a line \(L\), and have position vectors \(\left( {\begin{array}{*{20}{c}} { - 3} \\ { - 2} \\ 2 \end{array}} \right)\) and \(\left( {\begin{array}{*{20}{c}} 6 \\ 4 \\ { - 1} \end{array}} \right)\) respectively. Let O be the origin. This is shown on the following diagram.
The point C also lies on \(L\), such that \(\overrightarrow {{\text{AC}}} = 2\overrightarrow {{\text{CB}}} \).
Let \(\theta \) be the angle between \(\overrightarrow {{\text{AB}}} \) and \(\overrightarrow {{\text{OC}}} \).
Let D be a point such that \(\overrightarrow {{\text{OD}}} = k\overrightarrow {{\text{OC}}} \), where \(k > 1\). Let E be a point on \(L\) such that \({\rm{C\hat ED}}\) is a right angle. This is shown on the following diagram.
Find \(\overrightarrow {{\text{AB}}} \).
Show that \(\overrightarrow {{\text{OC}}} = \left( {\begin{array}{*{20}{c}} 3 \\ 2 \\ 0 \end{array}} \right)\).
Find \(\theta \).
(i) Show that \(\left| {\overrightarrow {{\text{DE}}} } \right| = (k - 1)\left| {\overrightarrow {{\text{OC}}} } \right|\sin \theta \).
(ii) The distance from D to line \(L\) is less than 3 units. Find the possible values of \(k\).
Line \({L_1}\) passes through points \({\text{A}}(1{\text{, }} - 1{\text{, }}4)\) and \({\text{B}}(2{\text{, }} - 2{\text{, }}5)\) .
Line \({L_2}\) has equation \({\boldsymbol{r}} = \left( \begin{array}{l}
2\\
4\\
7
\end{array} \right) + s\left( \begin{array}{l}
2\\
1\\
3
\end{array} \right)\) .
Find \(\overrightarrow {{\rm{AB}}} \) .
Find an equation for \({L_1}\) in the form \({\boldsymbol{r}} = {\boldsymbol{a}} + t{\boldsymbol{b}}\) .
Find the angle between \({L_1}\) and \({L_2}\) .
The lines \({L_1}\) and \({L_2}\) intersect at point C. Find the coordinates of C.
The diagram shows a parallelogram ABCD.
The coordinates of A, B and D are A(1, 2, 3) , B(6, 4,4 ) and D(2, 5, 5) .
(i) Show that \(\overrightarrow {{\rm{AB}}} = \left( {\begin{array}{*{20}{c}}
5\\
2\\
1
\end{array}} \right)\) .
(ii) Find \(\overrightarrow {{\rm{AD}}} \) .
(iii) Hence show that \(\overrightarrow {{\rm{AC}}} = \left( {\begin{array}{*{20}{c}}
6\\
5\\
3
\end{array}} \right)\) .
Find the coordinates of point C.
(i) Find \(\overrightarrow {{\rm{AB}}} \bullet \overrightarrow {{\rm{AD}}} \).
(ii) Hence find angle A.
Hence, or otherwise, find the area of the parallelogram.
The following diagram shows two perpendicular vectors u and v.
Let \(w = u - v\). Represent \(w\) on the diagram above.
Given that \(u = \left( \begin{array}{c}3\\2\\1\end{array} \right)\) and \(v = \left( \begin{array}{c}5\\n\\3\end{array} \right)\), where \(n \in \mathbb{Z}\), find \(n\).
Consider the lines \({L_1}\) , \({L_2}\) , \({L_2}\) , and \({L_4}\) , with respective equations.
\({L_1}\) : \(\left( {\begin{array}{*{20}{c}}
x\\
y\\
z
\end{array}} \right) = \left( {\begin{array}{*{20}{c}}
1\\
2\\
3
\end{array}} \right) + t\left( {\begin{array}{*{20}{c}}
3\\
{ - 2}\\
1
\end{array}} \right)\)
\({L_2}\) : \(\left( \begin{array}{l}
x\\
y\\
z
\end{array} \right) = \left( \begin{array}{l}
1\\
2\\
3
\end{array} \right) + p\left( \begin{array}{l}
3\\
2\\
1
\end{array} \right)\)
\({L_3}\) : \(\left( {\begin{array}{*{20}{c}}
x\\
y\\
z
\end{array}} \right) = \left( {\begin{array}{*{20}{c}}
0\\
1\\
0
\end{array}} \right) + s\left( {\begin{array}{*{20}{c}}
{ - 1}\\
2\\
{ - a}
\end{array}} \right)\)
\({L_4}\) : \(\left( {\begin{array}{*{20}{c}}
x\\
y\\
z
\end{array}} \right) = q\left( {\begin{array}{*{20}{c}}
{ - 6}\\
4\\
{ - 2}
\end{array}} \right)\)
Write down the line that is parallel to \({L_4}\) .
Write down the position vector of the point of intersection of \({L_1}\) and \({L_2}\) .
Given that \({L_1}\) is perpendicular to \({L_3}\) , find the value of a .
Two points P and Q have coordinates (3, 2, 5) and (7, 4, 9) respectively.
Let \({\mathop {{\text{PR}}}\limits^ \to }\) = 6i − j + 3k.
Find \(\mathop {{\text{PQ}}}\limits^ \to \).
Find \(\left| {\mathop {{\text{PQ}}}\limits^ \to } \right|\).
Find the angle between PQ and PR.
Find the area of triangle PQR.
Hence or otherwise find the shortest distance from R to the line through P and Q.
The line L1 is represented by \({{\boldsymbol{r}}_1} = \left( {\begin{array}{*{20}{c}}
2\\
5\\
3
\end{array}} \right) + s\left( {\begin{array}{*{20}{c}}
1\\
2\\
3
\end{array}} \right)\) and the line L2 by \({{\boldsymbol{r}}_2} = \left( {\begin{array}{*{20}{c}}
3\\
{ - 3}\\
8
\end{array}} \right) + t\left( {\begin{array}{*{20}{c}}
{ - 1}\\
3\\
{ - 4}
\end{array}} \right)\) .
The lines L1 and L2 intersect at point T. Find the coordinates of T.
Line \({L_1}\) has equation \({\boldsymbol{r}_1} = \left( {\begin{array}{*{20}{c}}
{10}\\
6\\
{ - 1}
\end{array}} \right) + s\left( {\begin{array}{*{20}{c}}
2\\
{ - 5}\\
{ - 2}
\end{array}} \right)\) and line \({L_2}\) has equation \({\boldsymbol{r}_2} = \left( {\begin{array}{*{20}{c}}
2\\
1\\
{ - 3}
\end{array}} \right) + t\left( {\begin{array}{*{20}{c}}
3\\
5\\
2
\end{array}} \right)\) .
Lines \({L_1}\) and \({L_2}\) intersect at point A. Find the coordinates of A.