3.10 Vector Equations of Lines

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

Find the equation of the line that is normal 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 are skew.

Consider the lines and defined by the equations  and  

Given that and  are coincident, find the value of .

Find the value of .

Two ships and  are travelling so that their position relative to a fixed point  at time , in hours, can be defined by the position vectors and  

The unit vectors  and are a displacement of 1 km due East and North of  respectively. 

Find the coordinates of the initial position of the two ships.

Show that the two ships will collide and find the time at which this will occur.

Find the coordinates of the point of collision.

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 .

The point  is the midpoint of . The line passing through points  and  can be defined by 

Show that the line intersects  at the point 

A car, moving at constant speed, takes 4 minutes to drive in a straight line from point  to point . 

At time , in minutes, the position vector of the car relative to the origin can be given in the form.

Find the vectors and 

A cat has decided to take a nap at point  

Show that the cat does not lie on the route along which the car drives.

Find the shortest distance between the car and the cat during the movement of the car.

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 rad, 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 Cartesian 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 .

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

A third line  is defined by the equations .

Determine the relationship between lines  and .

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.

Two drones X and Y are being flown over an area of rainforest to look for signs of illegal logging. Their positions relative to the observation centre, are given by

  and 

at time  minutes after take-off, . All distances are in metres.

Verify that the two drones will not collide.

Find the shortest distance between the two drones and the time at which it occurs.

A third drone Z begins its flight at  and its position relative to the observation centre is given by  

Each drone can observe a circular area of ground,   such that  where  is the height of the drone above the ground in metres.

Show that the area of ground that can be observed by drone Z five minutes after it takes off overlaps with the area of ground that can be observed by drone Y at that time.

Consider the tetrahedron ABCD, where A(3, 5, 8), B(-2, 3, 2), C(5, -1, 3) and D(-3, 0, 1). M is the midpoint of the line BC and point P lies along the line DM.

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

X is the midpoint of [AD].

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

A car is moving at a constant speed of 15 ms^-1 in the direction parallel to the vector  Two birds are perched at points  and . 

At , the car is located at  and the bird at point A starts to fly at a constant velocity of   ms^-1. The bird at point B begins to fly at a constant velocity in the direction of the vector  when . 

When bird A reaches the position of , both birds and the car lie in a straight line.

Find the equation of the line along which the birds and car lie.

Find the speed at which bird B is travelling.