IBMaths: AI HLDPExam Questions3. Geometry & Trigonometry3.7 Vector Properties

Vector Properties (DP IB Maths: AI HL)

Exam Questions

4 hours33 questions
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1a
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Show that the vectors a equals 2 i minus 6 j plus k and b equals negative i plus 3 j minus k are not parallel.

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1b
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Show that vertical line bold a bold times bold b vertical line less than vertical line bold a vertical line vertical line bold b vertical line

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1c
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Show that bold a bold times bold a equals vertical line bold a vertical line squared

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2
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Consider the two vectors bold s equals 3 i plus 4 j minus k and bold t equals negative 2 i plus 2 j minus 3 k. 

(i)
Find the cross product of bold s and bold t. 

(ii)
Hence, find the angle between bold s and bold t. Give your answer in radians.

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3
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The vectors bold a and bold b are defined by bold a bold space equals open parentheses table row 1 row cell negative 3 end cell row 1 end table close parentheses comma space bold b bold space equals open parentheses table row 5 row 2 row cell negative 2 end cell end table close parentheses. 

By finding the scalar product of bold a and bold b, find the angle between them. Give your answer in degrees.

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4a
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Let bold v space equals space open parentheses table row t row cell negative 3 end cell row cell t plus 2 end cell end table close parentheses and bold w equals open parentheses table row cell negative 6 end cell row 7 row t end table close parentheses. 

Given that bold v and bold w are perpendicular, find all possible values of t.

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4b
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Show that the angle between bold v and bold w is acute for all t greater than 7.

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5
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Consider the vectors bold a equals 3 i minus j plus 4 k and bold b equals left parenthesis 2 plus t right parenthesis i minus 2 j plus 2 t k..

Given that a and b are parallel and hence the vector product is equal to zero, determine the value of t.

 

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6a
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Consider the vectors bold a equals negative 2 i minus j plus 3 k and bold b equals 3 i plus 5 k. 

Find a vector of length 7 that is parallel to bold a.

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6b
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Find the vector that is normal to both bold a and bold b.

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7a
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A particle is subjected to a force of 36 N acting at an angle of 25 degree above the horizontal and a second force F at an angle of 17 degree  below the horizontal. There is also a resistive force of 50 N acting horizontally on the particle. This information can be seen in the diagram below.

q7-vector-properties-medium-ib-ai-hl

Given that the resultant horizontal force acting on the particle is 0 N, find the value of F.

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7b
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Show that the vertical component of the resultant force is 9.9 N.

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8a
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Consider the vectors bold r equals open parentheses table row 2 row 4 row cell negative 1 end cell end table close parentheses and bold t equals open parentheses table row cell negative 3 end cell row 5 row 3 end table close parentheses.

Show that 3 bold r cross times bold t equals 3 left parenthesis bold r cross times bold t right parenthesis.

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8b
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Find the area of a triangle which has vectors 3 bold r and bold t as two of its sides.

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9a
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On a calm day, a remote-controlled boat is being driven along a vector bold u equals i plus 3 j from one side of a pond to the other. 

The boat is retrieved and taken to the same starting point, to make the journey again but this time a steady wind causes the boat to travel in a direction represented by the vector bold w equals 2 i minus j. 

Calculate the angle, in degrees, between the direction of travel on its initial journey and the direction on its subsequent journey.

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9b
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During the first journey, the boat takes 6.3 seconds to travel the 7.56 m to the other side of the pond. 

Find the velocity vector of the boat.

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9c
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Given that during the second journey the boat covers a distance of 5.1 m, find the distance between the end points for both journeys.

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10a
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ABCD is a parallelogram with vertices straight A left parenthesis 2 comma space 3 comma space 0 right parenthesis comma space straight B left parenthesis 3 comma space 9 comma space 4 right parenthesis comma space straight C left parenthesis 7 comma space 4 comma space 2 right parenthesis and straight D left parenthesis 6 comma negative 2 comma negative 2 right parenthesis. 

Find the vectors AB with rightwards arrow on top and  AD with rightwards arrow on top.

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10b
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Find the area of the parallelogram.

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10c
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By finding the scalar product of BA with rightwards arrow on top and BC with rightwards arrow on top, determine if the angle straight A straight B with hat on top straight C is  acute or obtuse.

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11a
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The velocity of a river can be described by the vector bold a equals 2 i minus 3 j space kmh to the power of negative 1 end exponent  and a swimmer moves through the river with velocity bold b equals negative 4 i plus j kmh to the power of negative 1 end exponent  .

Find the speed at which the river is flowing and the swimmer is swimming.

 

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11b
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Find the resultant vector of the swimmer and the river.

 

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11c
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Find the bearing along which the swimmer actually moves.

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11d
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The swimmer is attempting to complete a 5 km race for charity. Given that the velocity vectors for the river and the swimmer do not change, determine how long it will take the swimmer to complete the challenge.

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1a
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Given open vertical bar a close vertical bar equals 5, b equals open parentheses table row 12 row cell negative 5 end cell end table close parentheses  and a times b equals 16  ,  find the angle between a and b.

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1b
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Consider a third vector c, where open vertical bar c close vertical bar equals 8.

When the angle between a and c is  pi over 4 , 

(i)
verify that open vertical bar a close vertical bar open vertical bar c close vertical bar less than 2 open vertical bar a times c close vertical bar comma
(ii)
find the component of vector c acting in the direction of vector a.

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2a
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The points straight A and straight B  have position vectors a  and b respectively.

 open vertical bar bold italic a close vertical bar equals 9 comma space bold italic a bold cross times bold italic b equals open parentheses table row 5 row cell negative 3 end cell row 2 end table close parentheses comma space bold italic a bold times bold italic b equals 16 

Find the angle between bold italic a and bold italic b.

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2b
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The points straight A and straight B form a triangle with the origin straight O.

Find open vertical bar bold italic b close vertical bar.

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3
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 ABCD is a parallelogram with vertices straight A open parentheses negative 2 comma 5 comma 3 close parentheses comma straight B comma straight C and straight D left parenthesis 3 comma negative 1 comma t right parenthesis where t greater than 0 .

 AB with rightwards arrow on top equals DC with rightwards arrow on top equals open parentheses table row 2 row 4 row 1 end table close parentheses 

Given that the area of the parallelogram is square root of 1221 units, find the value of t.

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4
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Two points straight A and straight B  have position vectors a equals open parentheses table row 6 row 2 row cell negative 5 end cell end table close parentheses  and b equals lambda open parentheses table row 3 row cell negative 2 end cell row cell negative 1 end cell end table close parentheses respectively. 

A third point C is located such that AC with rightwards arrow on top equals open parentheses table row 8 row cell negative 2 end cell row 0 end table close parentheses. 

Given that the angle between the vectors AB with rightwards arrow on top  and AC with rightwards arrow on top is obtuse, find the range of possible values for lambda.

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5a
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In vector form, the two parallel sides of a trapezium are given by bold italic u equals open parentheses table row t row cell 2 t minus 6 end cell row 1 end table close parentheses  and begin mathsize 16px style v equals open parentheses table row cell t minus 6 end cell row 4 row cell t minus 4 end cell end table close parentheses end style. Additionally, open vertical bar bold italic u close vertical bar open vertical bar bold italic v close vertical bar equals 18.

Given that t is an integer, find the value of t.

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5b
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A third side of the trapezium, with vector begin mathsize 16px style bold italic w equals open parentheses table row x row 3 row z end table close parentheses end style, is perpendicular to both bold italic u and bold italic v.

Given that open vertical bar bold italic w close vertical bar equals square root of 26, and that x is an integer, find the values of x and z.

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6a
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The points straight A left parenthesis negative 2 comma 5 comma 3 right parenthesis comma straight B comma straight C space and space straight D form a parallelogram.

AB with rightwards arrow on top equals open parentheses table row 6 row 2 row cell negative 1 end cell end table close parentheses, BC with rightwards arrow on top equals open parentheses table row cell negative 4 end cell row 3 row 4 end table close parentheses 

Find the area of the parallelogram.

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6b
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Show that the diagonals of the parallelogram are perpendicular to each another.

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6c
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Determine the nature of angle straight C straight D with hat on top straight A .

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7a
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ABCDEFGH is a cuboid as shown in the diagram below.

Point A is located at open parentheses 5 comma negative 3 comma space 4 close parentheses comma space space stack A B with rightwards arrow on top space equals 2 straight j space minus straight k and stack BC space with rightwards arrow on top equals 4 straight i plus straight j plus 2 straight k.

q7_3-7_vector-properties-_hard_ib-ai-hl_diagram

The perpendicular distance between the faces ABCD and EFGH of the cuboid is 2 square root of 105  units.

Find the coordinates of the point straight E open parentheses x comma y comma z close parentheses, where x comma space y comma space z element of straight integer numbers to the power of plus.

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7b
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A triangle is formed inside the cuboid by connecting the vertices B, C and E, where straight B straight C with hat on top straight E space equals theta. 

Using vector methods, find cos space theta .

 

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8
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The diagram below shows three forces acting on a particle.

q8_3-7_vector-properties-_hard_ib-ai-hl_diagram

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

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9a
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Point straight A  has position vector a equals 3 straight i plus 2 straight j minus straight k and point B has position vector b equals straight i minus straight j plus 4 straight k  relative to the origin straight O.

Find the area of the triangle straight A straight O with hat on top straight B .

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9b
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Point straight X  is located a distance of 8 units from the origin in the direction perpendicular to the plane formed by AOB.

Find all possible vectors OX with rightwards arrow on top .

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9c
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Find the volume of the tetrahedron AOBX . Give your answer in the form c square root of d, where c comma d element of straight integer numbers .

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10a
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 ABCDEFGH are vertices of a parallelepiped with the vectors AB with rightwards arrow on top comma space stack AD space with rightwards arrow on top space and space AE with rightwards arrow on top defined as p comma qand r respectively. theta is the angle between AE with rightwards arrow on top  and the normal to the base ABCD. This information can be seen in the diagram below.

q10_ib-aa-hl_vector-properties_diagram

Find an expression for

(i)
the area of the base ABCD,
(ii)
the perpendicular height of the parallelepiped.

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10b
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Hence, show that the volume of a parallelepiped is given by open vertical bar open parentheses p cross times q close parentheses times r close vertical barunits3.

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10c
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Find the volume of a parallelepiped with vertices straight A left parenthesis 5 comma 7 comma negative 3 right parenthesis comma space straight B left parenthesis 6 comma 10 comma negative 2 right parenthesis comma space straight C left parenthesis 9 comma 11 comma 0 right parenthesis space and space straight E left parenthesis 4 comma 5 comma 4 right parenthesis.

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11a
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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 v subscript 1 equals 2 straight i minus straight j plus 3 straight k space ms to the power of negative 1 end exponent.  At the same time a truck leaves a point O and moves along the ground below the hot air balloon at a velocity of 7 space m s to the power of negative 1 end exponent along a bearing of 079 degree. 

All distances are measured in metres and time in seconds. The base vectors straight i and text j end text  represent due east and due north respectively and the base vector straight k points upwards.

Write down

(i)
the speed at which the hot air balloon is travelling,
(ii)
the position vector of the hot air balloon five minutes after it changed direction,
(iii)
the velocity of the truck in base vector form.

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11b
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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 v subscript 2 equals 2 straight i minus straight j space ms to the power of negative 1 end exponent. 

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

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1a
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Given open vertical bar bold italic a close vertical bar equals 6 comma space open vertical bar bold italic b close vertical bar equals 2 and bold italic a bold cross times bold italic b equals open parentheses table row cell negative 5 end cell row 2 row 7 end table close parentheses find the possible values of  bold italic a bold colon bold italic b .

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1b
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nsider a third vector c, where open vertical bar c close vertical bar equals 5.

Given that the angle between a and c is  straight pi over 3,

(i)
find open vertical bar a times c close vertical bar and open vertical bar a cross times c close vertical bar,
(ii)
find the component of vector c acting perpendicular to vector a.

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2a
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The points A and B have position vectors a and b respectively.

open vertical bar a close vertical bar equals square root of 35 comma space b equals 2 straight i plus 2 straight j plus straight k comma space a cross times b equals open parentheses table row 7 row cell negative 11 end cell row 8 end table close parentheses and theta  is the angle between a and b.

Find cos space theta .

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2b
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A third point C is located such that its position vector  c equals open parentheses table row x row y row z end table close parentheses

Given that c equals n open parentheses a cross times b close parentheses  find the unit vector in the direction of c.

 

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3
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ABCDE is a pentagon, where  straight A open parentheses 0 comma 4 comma negative 1 close parentheses comma space straight D left parenthesis negative 1 comma 7 comma 3 right parenthesis comma space AB with rightwards arrow on top equals open parentheses table row cell negative 4 end cell row 2 row 5 end table close parentheses and CD with rightwards arrow on top equals open parentheses table row 1 row cell negative 1 end cell row cell negative 1 end cell end table close parentheses .

Given that BD with rightwards arrow on top equals AE with rightwards arrow on top find the area of triangle straight B straight C with hat on top straight D as a percentage of the total area of the pentagon.

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4a
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Three points, straight A left parenthesis negative 3 comma negative 1 right parenthesis, straight B left parenthesis 2 comma 2 right parenthesis  and C, are located on a straight line where AC with rightwards arrow on top equals lambda AB with rightwards arrow on top. A fourth point straight D, is located such that DB with rightwards arrow on top is perpendicular to OB with rightwards arrow on top and open vertical bar DB close vertical bar equals 2 square root of 22 .

Find straight D straight B with hat on top straight C.

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4b
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Given that the area of the triangle BCD equals 53.1units2 correct to 3 significant figures, find lambda.

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5
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Consider a regular hexagon ABCDEF with sides of length 9 square root of 2units. The position vectors of straight A and straight E are a equals negative 11 straight i minus 4 straight j plus 5 straight k  and e equals 3 straight i minus 3 straight j minus 12 straight k respectively.

Given that the coordinates of straight F are open parentheses r comma t comma t close parentheses,  where r comma t element of straight integer numbers comma space r not equal to 0. find the value of r and t.

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6a
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ABCD is a parallelogram defined by the vectors AB with rightwards arrow on top equals p and AD with rightwards arrow on top equals q, where p equals open parentheses table row 1 row 5 end table close parentheses and q equals open parentheses table row cell 2 t plus 2 end cell row t end table close parentheses. 

Given that the angle straight B straight A with hat on top straight D is acute, find the range of values for t.

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6b
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AB with rightwards arrow on top is enlarged by a factor of k.

Show that k p times q equals k open parentheses p times q close parentheses.

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6c
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Given that k equals 7, find the range of possible values for the area of the enlarged parallelogram.

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7
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Consider the cuboid ABCDEFGH as shown in the diagram below. The position vectors of straight A comma space straight C comma space straight D and straight E  are  a equals negative 5 straight i plus 2 straight j minus 4 straight k, c equals 2 straight i plus 3 straight j plus straight k, d equals negative straight i plus straight j and e equals negative 3 straight i plus straight j plus 2 straight k respectively.

straight X is a point located on the line open square brackets EC close square brackets such that EX with rightwards arrow on top equals straight lambda EC with rightwards arrow on top.

q7_ib-aa-hl_vector-properties_very_hard_diagram

Find the shortest length open vertical bar HX close vertical bar .

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8
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The following force diagram shows three forces acting on a particle:

q8_3-7_vector-properties-_very-hard_ib-ai-hl_diagram

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

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9a
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Consider a tetrahedron ABCV where AB with rightwards arrow on top equals p, AC with rightwards arrow on top equals q and AV with rightwards arrow on top equals r. The perpendicular height, h , of the tetrahedron from the base ABC makes an angle of theta with r.

This information is shown in the diagram below.

q9_ib-aa-hl_vector-properties_very-hard_diagram

Find an expression for the volume of the tetrahedron in terms of p comma q space a n d space r.

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9b
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Find the volume of the tetrahedron when p equals open parentheses table row 4 row 5 row cell negative 2 end cell end table close parentheses, q equals open parentheses table row 2 row 2 row cell negative 5 end cell end table close parentheses, r equals open parentheses table row 1 row 3 row 4 end table close parentheses

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9c
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Hence find the shortest distance between vertex A and its opposite face.

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10
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Consider a parallelepiped ABCDEFGH  with vertices straight A open parentheses 2 comma 5 comma negative 1 close parentheses comma space straight B open parentheses 3 comma 8 comma 0 close parentheses comma straight C open parentheses 5 comma 4 comma 3 close parentheses and straight H left parenthesis 5 comma 2 comma 7 right parenthesis  as seen in the diagram below.

q10_ib-aa-hl_vector-properties_very_hard_diagram

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

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11a
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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 4 space kmh to the power of negative 1 end exponentalong a bearing of 229 degree. The river flows in the direction given by the vector open parentheses table row 1 row cell negative 5 end cell end table close parentheses. 

The base vectors straight i space and space straight j represent due east and due north respectively. 

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

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11b
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Find the component of the explorer’s velocity in the direction of the underground river.

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11c
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After they have travelled for 2.5 hours, the explorer finds the treasure.

Find the perpendicular distance between the location of the treasure and the underground river.

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