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AHL 3.15—Classification of lines

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Topic 3— Geometry and trigonometry » AHL 3.15—Classification of lines

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Directly related questions

21M.1.AHL.TZ1.11a.i:
Show that the point $(- 1, 0, 3)$ lies on $L_{1}$ .
21M.1.AHL.TZ1.11a.ii:
Find a vector equation of $L_{1}$ .
21M.1.AHL.TZ1.11b:
Find the possible values of $a$ when the acute angle between $L_{1}$ and $L_{2}$ is $45 °$ .
21M.1.AHL.TZ1.11c:
It is given that the lines $L_{1}$ and $L_{2}$ have a unique point of intersection, $A$ , when $a \neq k$ .

Find the value of $k$ , and find the coordinates of the point $A$ in terms of $a$ .
21M.1.AHL.TZ2.8a:
Show that $l_{1}$ and $l_{2}$ do not intersect.
21M.1.AHL.TZ2.8b:
Find the minimum distance between $l_{1}$ and $l_{2}$ .
22M.2.AHL.TZ2.11d.i:
Find the coordinates of $P$ .
22M.2.AHL.TZ2.11d.ii:
Determine the length of time between the first airplane arriving at $P$ and the second airplane arriving at $P$ .
22M.1.AHL.TZ1.11c:
Find the distance between $L$ and $\prod_{3}$ .
22M.1.AHL.TZ1.11a:
Show that the three planes do not intersect.
22M.1.AHL.TZ1.11b.ii:
Find a vector equation of $L$ , the line of intersection of $\prod_{1}$ and $\prod_{2}$ .
22M.1.AHL.TZ1.11b.i:
Verify that the point $P (1, - 2, 0)$ lies on both $\prod_{1}$ and $\prod_{2}$ .
17M.2.AHL.TZ2.H_9a:
Find the vector equation of the line (BC).
17M.2.AHL.TZ2.H_9b:
Determine whether or not the lines (OA) and (BC) intersect.
17M.2.AHL.TZ2.H_9c:
Find the Cartesian equation of the plane Π $_1$ , which passes through C and is perpendicular to $\overrightarrow {{\rm{OA}}}$ .
17M.2.AHL.TZ2.H_9d:
Show that the line (BC) lies in the plane Π $_1$ .
17M.2.AHL.TZ2.H_9e:
Verify that 2j + k is perpendicular to the plane Π $_2$ .
17M.2.AHL.TZ2.H_9f:
Find a vector perpendicular to the plane Π $_3$ .
17M.2.AHL.TZ2.H_9g:
Find the acute angle between the planes Π $_2$ and Π $_3$ .
18M.2.AHL.TZ1.H_11a:
Show that the two submarines would collide at a point P and write down the coordinates of P.
18M.2.AHL.TZ1.H_11b.i:
Show that submarine B travels in the same direction as originally planned.
18M.2.AHL.TZ1.H_11b.ii:
Find the value of t when submarine B passes through P.
18M.2.AHL.TZ1.H_11c.i:
Find an expression for the distance between the two submarines in terms of t.
18M.2.AHL.TZ1.H_11c.ii:
Find the value of t when the two submarines are closest together.
18M.2.AHL.TZ1.H_11c.iii:
Find the distance between the two submarines at this time.
17M.1.SL.TZ1.S_8a.i:
Find $\overrightarrow {AB}$ .
17M.1.SL.TZ1.S_8a.ii:
Hence, write down a vector equation for ${L_1}$ .
17M.1.SL.TZ1.S_8b:
A second line ${L_2}$ , has equation r = $\left( {\begin{array}{*{20}{c}} 1 \\ {13} \\ { - 14} \end{array}} \right) + s\left( {\begin{array}{*{20}{c}} p \\ 0 \\ 1 \end{array}} \right)$ .

Given that ${L_1}$ and ${L_2}$ are perpendicular, show that $p = 2$ .
17M.1.SL.TZ1.S_8c:
The lines ${L_1}$ and ${L_1}$ intersect at $C(9,{\text{ }}13,{\text{ }}z)$ . Find $z$ .
17M.1.SL.TZ1.S_8d.i:
Find a unit vector in the direction of ${L_2}$ .
17M.1.SL.TZ1.S_8d.ii:
Hence or otherwise, find one point on ${L_2}$ which is $\sqrt 5$ units from C.

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