Show that the vectors and
are not parallel.
Show that
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Select a download format for 3.9 Vector Properties
Show that the vectors and
are not parallel.
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Show that
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Show that
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Consider the two vectors and
.
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The vectors and
are defined by
By finding the scalar product of and
, find the angle between them. Give your answer in degrees.
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Let and
.
Given that and
are perpendicular, find all possible values of
.
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Show that the angle between and
is acute for all
.
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Consider the vectors and
By finding the vector product, determine the value of given that
and
are parallel.
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Consider the vectors and
Find a vector of length 7 that is parallel to .
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Find the vector that is normal to both and
.
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Given the vectors
and
, show that
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Given any two non-zero vectors and
, show that
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Consider the vectors and
.
Show that .
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Find the area of a triangle which has vectors and
as two of its sides.
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On a calm day, a remote-controlled boat is being driven along a vector 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 .
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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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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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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is a parallelogram with vertices
and
Find the vectors and
.
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Find the area of the parallelogram.
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By finding the scalar product of and
, determine if the angle
is acute or obtuse.
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Given and
, find the angle between
and
.
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Consider a third vector c, where .
When the angle between and
is
, show that
.
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The points and
have position vectors
and
respectively.
Find the angle between and
.
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The points and
form a triangle with the origin
.
Find .
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is a parallelogram with vertices
and
where
.
Given that the area of the parallelogram is units, find the value of t.
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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
.
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In vector form, the two parallel sides of a trapezium are given by and
. Additionally,
.
Given that is an integer, find the value of
.
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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
.
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The points form a parallelogram.
,
Find the area of the parallelogram.
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Show that the diagonals of the parallelogram are perpendicular to each another.
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Determine the nature of angle .
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is a cuboid as shown in the diagram below.
Point A is located at .
The perpendicular distance between the faces ABCD and EFGH of the cuboid is units.
Find the coordinates of the point , where
.
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A triangle is formed inside the cuboid by connecting the vertices and
, where
.
Using vector methods, find .
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The points form the vertices of a parallelogram with position vectors
respectively.
Show that the area of the parallelogram is .
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Hence show that the shortest distance from to
is
.
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Point has position vector
and point B has position vector
relative to the origin
.
Find the area of the triangle .
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Point is located a distance of 8 units from the origin in the direction perpendicular to the plane formed by
.
Find all possible vectors .
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Find the volume of the tetrahedron . Give your answer in the form
, where
.
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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
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Hence, show that the volume of a parallelepiped is given by units3.
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Find the volume of a parallelepiped with vertices .
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Given and
, find the possible values of
.
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Consider a third vector c, where
Given that the angle between and
is
, find
.
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The points and
have position vectors
and
respectively.
and
is the angle between
and
.
Find .
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A third point is located such that its position vector
Show that .
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is a pentagon, where
and
.
Given that find the area of triangle
as a percentage of the total area of the pentagon.
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Three points, ,
and C, are located on a straight line where
. A fourth point
, is located such that
is perpendicular to
and
.
Find .
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Given that the area of the triangle units2 correct to 3 significant figures, find
.
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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
.
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ABCD is a parallelogram defined by the vectors and
, where
and
.
Given that the angle is acute, find the range of values for
.
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is enlarged by a factor of
.
Show that .
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Given that , find the range of possible values for the area of the enlarged parallelogram.
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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 .
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Show that for any two vectors and
,
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Hence show that when .
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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
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Find the volume of the tetrahedron when ,
,
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Hence find the shortest distance between vertex A and its opposite face.
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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.
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