Further Vectors (Edexcel International A Level Maths: Pure 4)

Find the equation of each of these lines in vector form.

The line joining  to .

The line passing through  in the same direction as .

Find the equation of each of these lines in vector form.

Through in the direction 

Through and . 

Show that the point  lies on the line found in part (a) (ii).

 and  are the points on the line    with  and  respectively. 

Find the position vectors  and .

Find .

The coordinates of three points are and  .

Find  and  .

Calculate  and  ,  and the scalar product  .

Hence, find the angle between the vectors  and . 
Give your answer in degrees correct to 1 decimal place.

The vertices of triangle  are the points with coordinates  and  .

Show that .

What does the result of part (i) tell you about the triangle ?

Two lines  and have equations  and  respectively.

Show that the two lines meet and find the position vector of the point of intersection.

Two lines  and  have equations  and  respectively.

The line  has equation 

Point  is the point on line  such that the line connecting and the point  is perpendicular to .

Find the position vector, , of point .

Hence find the shortest distance from point  to the line .

The following diagram depicts a large bank vault, the walls, floor and ceiling of which are all perfectly rectangular.

The corner  is taken to be the origin of the coordinate system, and the units on the diagram are all given in metres.

  is the diagonal running across the room from corner  to corner .

By first finding the vector ,  write a vector equation of the line that passes through  and .

A motion detector is located at point , and is set to sound an alarm if anything moves within 5 metres of it.

If a small insect flies in a straight line from point  to point , will the alarm be triggered?  You must provide clear mathematical workings to justify your answer.

Find the equation of each of these lines in vector form.

The line joining  to .

The line passing through   parallel to .

Find the equation of each of these lines in vector form.

Through in the same direction as .

Through and . 

Show that the point  does not lie on the line found in part (a) (ii).

 and are the points on the line  with  and  respectively.  Given that the distance from  to  is 9 units, find the possible values of .

The coordinates of three points are  and  .

Find  and .

By considering the scalar product , or otherwise, calculate the angle between  and  .  Give your answer in degrees, accurate to 1 decimal place.

The vertices of triangle  are the points with coordinates  and  .

Calculate the scalar products  and 

What does the result of part (i) tell you about the triangle ?

Determine whether each of the following pairs of lines intersect, are parallel, or are skew.  If the lines intersect, find the coordinates of the point of intersection.

   and    

     and    .

Find the perpendicular distance of the point   to the line .

In the magical kingdom of Cartesia, all positions are measured relative to the ancient stone of power known as the Origin.  This reference system corresponds to the standard  coordinate system used in mathematics, as shown in the diagram below:

Prince Vector, son of King Prime of Cartesia, needs to fly on his magical unicorn from the top of the Mystic Pedestal all the way to Cloud City, on an urgent rescue mission to save the kingdom from certain doom. 

The Mystic Pedestal is 14 miles west and 8 miles north of the Origin, and its top is one mile up from the level of the Origin.  Cloud City is 11 miles east and 13 miles north of the Origin, and it is 11 miles up from the level of the Origin.

Since there is not much time, the prince must fly directly from the top of the Mystic Pedestal to Cloud City.  Unfortunately, his unicorn’s magic levels are low.  In order for the unicorn to recharge it must pass within 12 miles of the Origin during the flight, and must do this before reaching the halfway point between the Mystic Pedestal and Cloud City.  If the unicorn does not recharge before this point, then it and the prince will crash into the barren wastes, and the kingdom will perish.

After the prince departs, his sister Hypatia (a keen mathematics student and heiress to the throne) remembers that there is a vector method that can be used to determine whether or not her brother’s mission will succeed.

Determine whether or not the prince will reach Cloud City successfully, using clear mathematical workings to justify your answer.

Given that the coordinates of and  are    and  respectively, find the equation of each of the following lines in vector form.

The line joining  to the midpoint of  .

The line passing through , parallel to the line in part (i).

Given that the coordinates of and are  and   respectively, find the equation of each of the following lines in vector form.

The line through and .

The line through , parallel to  .

Determine whether the point  lies on either of the lines found in part (a).

The coordinates of three points are  ,    and  .  

The point is the midpoint of , and the point lies on .

Given that , find the equation of the line through points and  in vector form.

The coordinates of three points are  .

Calculate the angle between and .  Give your answer in degrees, accurate to 1 decimal place.

The vertices of triangle are the points with coordinates 

Use a vector method to prove that  is a right-angled triangle.

Determine whether each of the following pairs of lines intersect, are parallel, or are skew.  For any lines that intersect, determine the point(s) of intersection.

and  

 and 

 and  

Find the coordinates of the point on the line  that is closest to the point    and hence determine the minimum distance from point  to the line.

The following diagram depicts imaginary lines connecting five points in space:

Points and are the locations, respectively, of the stars Arccirclus, Betacarotjuse, α-Capella and Denomineb.  Point is the location of the Stellamortis battle station, a planet-killing atrocity being built by the evil Galactic Imperium.  Coordinates are given relative to an origin point in accordance with the standard  coordinate system, and the units for all coordinates are parsecs.

The forces of the Star Rebellion are prepared to launch a strike to destroy the battle station, but they are unsure of its exact location.  According to data recovered from a smuggled droid, however, the following facts are known about the location of point :

• Point is in the First Octant of the galaxy, where and coordinates are all positive.

• The distance from point to point is exactly parsecs.

• Points and form the base of a pyramid, with its apex at point .

• The point on closest to point  is also the point where the two diagonals of the pyramid’s base intersect.

As the rebellion’s Chief Mathematician, it is your job to use the information provided to find the exact coordinates of point .  The fate of the galaxy is in your mathematical hands!