IBMaths: AA HLDPTopic Questions3. Geometry & Trigonometry3.9 Vector Properties

Vector Properties (DP IB Maths: AA HL)

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3.9 Vector Properties

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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. 

By finding the vector product, determine the value of t comma given that bold a and bold b are parallel.

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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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Given the vectors bold r equals negative i plus 2 j plus k comma bold s equals 5 i plus j minus k and bold t equals 2 i plus 2 j plus 4 k, show that

(i)
bold r times left parenthesis bold s plus bold t right parenthesis equals bold r bold times bold s plus bold r times bold t 

(ii)
bold r cross times left parenthesis bold s plus bold t right parenthesis equals bold r cross times bold s plus bold r cross times bold t

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7b
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Given any two non-zero vectors bold a and bold b, show that bold a cross times bold b equals negative bold b cross times bold a bold.

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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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1a
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Given open vertical bar a close vertical bar equals 5 comma space 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 bold italic a and bold italic b .

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1b
Sme Calculator3 marks

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, show that open vertical bar a times open parentheses b plus c close parentheses close vertical bar equals 16 plus 20 square root of 2.

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2a
Sme Calculator4 marks

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 minus 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
Sme Calculator4 marks

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 bold italic 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. 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
Sme Calculator4 marks

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
Sme Calculator3 marks

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 AB with rightwards arrow on top equals 2 straight j minus straight k space and space BC with rightwards arrow on top equals 4 straight i plus straight j plus 2 straight k.

q7_ib-aa-hl_vector-properties_hard_diagram

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

Find the coordinates of the point straight E left parenthesis x comma space y comma space z right parenthesis, where x comma space y comma space z element of straight integer numbers.

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7b
Sme Calculator3 marks

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

Using vector methods, find cos space theta.

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8a
Sme Calculator4 marks

The points straight A comma space straight B comma space straight C space and space straight D form the vertices of a parallelogram with position vectors a comma b comma c space and space d  respectively.

Show that the area of the parallelogram is open vertical bar a cross times b plus b cross times d plus d cross times a close vertical bar.

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8b
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Hence show that the shortest distance from straight B to AD is fraction numerator open vertical bar a cross times b plus b cross times d plus d cross times a close vertical bar over denominator open vertical bar d minus a close vertical bar end fraction .

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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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1a
Sme Calculator4 marks

Given open vertical bar a close vertical bar equals 6 comma space open vertical bar b close vertical bar equals 2 and a cross times 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 times bold italic b .

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1b
Sme Calculator3 marks

Consider a third vector c, where open vertical bar c close vertical bar equals 5.

Given that the angle between a and c is pi over 3, find open vertical bar a cross times open parentheses b plus c close parentheses close vertical bar .

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2a
Sme Calculator4 marks

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

 open vertical bar a close vertical bar equals square root of 35 comma b equals 2 i minus 2 j plus 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
Sme Calculator2 marks

A third point C is located such that its position vector  c equals open parentheses table row 28 row cell negative 44 end cell row 32 end table close parentheses

Show that c equals 4 a cross times b.

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3
Sme Calculator8 marks

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
Sme Calculator3 marks

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
Sme Calculator4 marks

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

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5
Sme Calculator9 marks

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
Sme Calculator3 marks

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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8a
Sme Calculator5 marks

Show that for any two vectors a and b,open vertical bar a cross times b close vertical bar squared minus open parentheses a times b close parentheses squared equals open vertical bar a close vertical bar squared open vertical bar b close vertical bar squared space cos space 2 theta

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8b
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Hence show that when sin space theta equals fraction numerator square root of 2 over denominator 2 end fraction comma open vertical bar a cross times b close vertical bar equals a times b.

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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
Sme Calculator3 marks

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
Sme Calculator4 marks

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

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10
Sme Calculator8 marks

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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