Vector Properties (DP IB Maths: AI HL)

Given ,  and   ,  find the angle between and .

Consider a third vector c, where 

When the angle between  and  is   , 

(i)

verify that

(ii)

find the component of vector  acting in the direction of vector 

The points  and  have position vectors  and  respectively.

Find the angle between and .

The points  and form a triangle with the origin .

Find .

  is a parallelogram with vertices  and where .

Given that the area of the parallelogram is units, find the value of t.

Two points  and  have position vectors  and  respectively. 

A third point  is located such that . 

Given that the angle between the vectors  and is obtuse, find the range of possible values for .

In vector form, the two parallel sides of a trapezium are given by   and . Additionally, .

Given that is an integer, find the value of .

A third side of the trapezium, with vector , is perpendicular to both  and .

Given that , and that is an integer, find the values of and .

The points  form a parallelogram.

,  

Find the area of the parallelogram.

Show that the diagonals of the parallelogram are perpendicular to each another.

Determine the nature of angle .

ABCDEFGH is a cuboid as shown in the diagram below.

Point A is located at  and 

The perpendicular distance between the faces ABCD and EFGH of the cuboid is  units.

Find the coordinates of the point , where .

A triangle is formed inside the cuboid by connecting the vertices B, C and E, where . 

Using vector methods, find .

The diagram below shows three forces acting on a particle.

Find the magnitude and direction of the resultant forces acting on the particle in the horizontal and vertical directions.

Point  has position vector  and point B has position vector  relative to the origin .

Find the area of the triangle .

Point  is located a distance of 8 units from the origin in the direction perpendicular to the plane formed by .

Find all possible vectors .

Find the volume of the tetrahedron . Give your answer in the form , where .

  are vertices of a parallelepiped with the vectors  defined as and  respectively. is the angle between  and the normal to the base ABCD. This information can be seen in the diagram below.

Find an expression for

Hence, show that the volume of a parallelepiped is given by units³.

Find the volume of a parallelepiped with vertices .

A hot air balloon travels vertically upwards from the point O on the ground for 12 metres and then moves in a straight line with constant velocity described by the vector .  At the same time a truck leaves a point O and moves along the ground below the hot air balloon at a velocity of along a bearing of . 

All distances are measured in metres and time in seconds. The base vectors and   represent due east and due north respectively and the base vector  points upwards.

Write down

When it reaches a vertical height of 0.92 km the hot air balloon changes direction again and moves with constant velocity described by the vector . 

Find the component of the balloon’s velocity in the direction of the road the truck had driven along.

Given and find the possible values of   .

nsider a third vector c, where .

Given that the angle between and  is  ,

(i)

find and ,

(ii)

find the component of vector acting perpendicular to vector .

The points A and B have position vectors  and respectively.

 and  is the angle between  and .

Find .

A third point C is located such that its position vector  

Given that   find the unit vector in the direction of c.

is a pentagon, where   and .

Given that find the area of triangle  as a percentage of the total area of the pentagon.

Three points, ,   and C, are located on a straight line where . A fourth point , is located such that is perpendicular to and .

Find .

Consider a regular hexagon  with sides of length units. The position vectors of  and are   and  respectively.

Given that the coordinates of  are ,  where  find the value of  and .

ABCD is a parallelogram defined by the vectors  and , where  and . 

Given that the angle is acute, find the range of values for .

 is enlarged by a factor of .

Show that .

Given that , find the range of possible values for the area of the enlarged parallelogram.

Consider the cuboid  as shown in the diagram below. The position vectors of and  are  , ,  and respectively.

 is a point located on the line  such that .

Find the shortest length .

The following force diagram shows three forces acting on a particle:

Given that the resultant force on the particle in the vertical direction is 14.195 N downwards, find the size of the angle .

Consider a tetrahedron  where , and . The perpendicular height, , of the tetrahedron from the base  makes an angle of  with .

This information is shown in the diagram below.

Find an expression for the volume of the tetrahedron in terms of

Find the volume of the tetrahedron when , , 

Hence find the shortest distance between vertex A and its opposite face.

Consider a parallelepiped  with vertices  and  as seen in the diagram below.

By first finding an expression for the perpendicular height of the object, find the volume of the parallelepiped.

An explorer is trying to find some treasure that they believe is hidden in a wild region near an underground river. They set off from a point O and travel at a velocity of along a bearing of . The river flows in the direction given by the vector . 

The base vectors  represent due east and due north respectively. 

Write down the velocity of the explorer as a column vector.

Find the component of the explorer’s velocity in the direction of the underground river.

After they have travelled for 2.5 hours, the explorer finds the treasure.