3.8 Vector Equations of Lines

Find the vector equations of a line that is parallel to the vector and passes through the point 

Find the equation of the line that is perpendicular to the vector and passes through the point , leaving your answer in the form where  and

Consider the two lines and  defined by the equations: 

Find the scalar product of the direction vectors.

Hence, find the angle, in radians, between the and .

Consider the lines  and defined by: 

Show that the lines are not parallel.

Hence, show that the lines and do not intersect.

Consider the line  which can be defined by both  and 

Find the value of .

Find the value of .

Consider the line , which can be represented by the equation  and , which can be represented by the equation

Write down the equation for  in its vector form.

Find vector product of the direction vectors of  and .

Hence find the angle between  and .

The lines  and can be defined by: 

Write down the parametric equations for .

Given that and intersect at point , 

(i)

find the value of . 

(ii)

determine the coordinates of the point of intersection, .

Consider the triangle . The points ,  and  have coordinates  and respectively.

M is the midpoint of  

Find the coordinates of the midpoint M.

Hence, find a vector equation of the line, , that passes through points and .

Show that the line  is perpendicular to [AB].

Hence calculate the area of the triangle ABC.

A line  has the equation  and intersects the line  with equation at point P, when .

A third line runs parallel to  and also intersects  at point .  

Find the parametric equations of .

Find the distance .

Consider the two intersecting lines  and  defined by the equations:

Given that the angle between  and  is 1.281 radians, correct to 4 significant figures, find the value of , where .

Find the value of , giving your answer correct to 3 significant figures.

Consider the two lines  and , where  passes through the points  and  and  is defined by the Parametric equations:  

Find the shortest distance between the two lines.

Consider the line  as defined by the equation . 

A point  lies at a distance of  units perpendicular from a point  on .

Find all possible coordinates of P.

Given that , write down the set of parametric equations that defines the line that passes through points P and X.

A wheelchair ramp is required to provide access to a building with a door that is located 22 cm above ground level.  The maximum angle that a ramp must be from the horizontal is 4.8°.

Calculate the minimum horizontal distance that the ramp must extend out.

The wheelchair ramp is supported by a steel frame.  A cross section of the ramp can be seen in the diagram below.  A metal strut joins M, the midpoint of [AC], to a point X on the line [AB]. [AB].XM=11.1 cm and =90°.  

Using the horizontal distance found in part (a) and assuming that point A is at the origin, use a vector method to calculate the length XB.

Some children are watching a canal boat navigating a system of locks. The boat starts at coordinates  relative to the point at which the children are standing.

The direction is due east, the direction is due north and the direction is vertically upwards. All distances are measured in metres and the children are taken to be standing at the origin.

The boat travels with direction vector  for 10 metres to get into the lock and then descends vertically downwards in the lock for 11 metres before continuing along the same direction vector as it was travelling along before entering the lock.

Find the coordinates of the entrance of the lock, given that the boat is now closer to the children.

Find the equation of the line along which the boat is travelling after it leaves the lock

On the next part of the journey at the point when the boat is closest to the children a child throws a flower to the boat driver. Given that the flower travels in a straight line and is caught by the boat driver, find the distance that the flower travelled.

Consider the tetrahedron ABCD, where , ,    and . M is the midpoint of the line  and point  lies along the line .

Given that the volume of the tetrahedron ABCP is  of the volume of the tetrahedron ABCD, find the Vector equation of the line going through points A and P.

 X is the midpoint of .

Find the coordinates of the point of intersection between the line found in part (a) and the line going through .

An adventure park structure is made out of steel rods arranged into a frame. As a part of the structure a red rod joins the coordinates  to  and a blue rod joins to .

Find the coordinates of the point where the red and blue rods meet each other.

The red rod also meets a yellow rod which has the vector equation . The point intersection of the red and blue rods and the red and yellow rods are joined by a taut rope.

Find the length of the rope.

A graphics designer joins the coordinates  to  and also plots the line with parametric equations:

Find a Vector equation of the line joining the points and and show that it does not intersect the line .

Find the two possible coordinates of the point on such that the angle is equal to   radians.