2000 Paper 2 Q7

Year: 2000
Paper: 2
Question Number: 7

Course: UFM Pure
Section: Vectors

Difficulty: 1600.0 Banger: 1486.1

vectors angle between lines scalar product trigonometric parametric circle line-plane intersection locus

Problem

The line \(l\) has vector equation \({\bf r} = \lambda {\bf s}\), where \[ {\bf s} = (\cos\theta+\sqrt3\,) \; {\bf i} +(\surd2\;\sin\theta)\;{\bf j} +(\cos\theta-\sqrt3\,)\;{\bf k} \] and \(\lambda\) is a scalar parameter. Find an expression for the angle between \(l\) and the line \mbox{\({\bf r} = \mu(a\, {\bf i} + b\,{\bf j} +c\, {\bf k})\)}. Show that there is a line \(m\) through the origin such that, whatever the value of \(\theta\), the acute angle between \(l\) and \(m\) is \(\pi/6\). A plane has equation \(x-z=4\sqrt3\). The line \(l\) meets this plane at \(P\). Show that, as \(\theta\) varies, \(P\) describes a circle, with its centre on \(m\). Find the radius of this circle.

No solution available for this problem.

Rating Information

Difficulty Rating: 1600.0

Difficulty Comparisons: 0

Banger Rating: 1486.1

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Problem source

The  line $l$ has vector equation ${\bf r} = \lambda {\bf s}$,
where 
\[
{\bf s} = (\cos\theta+\sqrt3\,) \; {\bf i} +(\surd2\;\sin\theta)\;{\bf j}
+(\cos\theta-\sqrt3\,)\;{\bf k} 
\]
and $\lambda$ is a scalar parameter. Find an expression for the 
angle between $l$ and the line 
\mbox{${\bf r} = \mu(a\, {\bf i} + b\,{\bf j} +c\,  {\bf k})$}.
Show that there is a line $m$ through the origin
such that, whatever the value of 
 $\theta$, the acute angle between $l$ and $m$ is $\pi/6$.

A plane has  equation  $x-z=4\sqrt3$. The line $l$ meets this plane 
at $P$. Show that, as $\theta$ varies, $P$ describes a circle,
with its centre on $m$. Find the radius of this circle.

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