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OEF Vectors 3D

OEF Vectors 3D
--- Introduction ---

This module contains actually 19 exercises on vectors in 3D
(linear combinations, angle, length, scalar product, vector product, cross
product, etc.).

Area of parallelogram

Compute the area of the parallelogram in the cartesian space whose 4 vertices are

(,,) , (,,) , (,,) , (,,) .

Area of triangle

Compute the area of the triangle in the cartesian space whose 3 vertices are

(,,) , (,,) , (,,) .

Angle

We have 3 points in the space:

,
,
.

Compute the angle
(in degrees, between 0 and 180).

Combination

Let

,
,

be three space vectors. Compute the vector

.

Combination 2 vectors

Let

,

be three space vectors. Compute the vector

.

Combination 4 vectors

Let

,
,
,

be four space vectors. Compute the vector

.

Find combination

Let

,
,

be three space vectors. Try to express
as a linear combination of v_{1},
and
.

Find combination 2 vectors

Let

,
,

be two space vectors. Try to express
as a linear combination of
and
.

Given scalar products

Let

,
,

be three space vectors. Find the vector
having the following scalar products:

,
,
.

Given vector product

Let
be a space vector. Determine the vector
such that the vector product
equals (,,).

Vector product and length

Let
be a space vector. We have another vector v which is perpendicular to
. Given that the length of
is equal to , what is the length of the vector product
?

Vector product and length II

Let
be a space vector. We have another vector
whose length is . Given that the scalar product
, what is the length of the vector product
?

Vertex of parallelogram

We have a parallelogram
in the cartesian space, whose 3 first vertices are at the coordinates

= (,,) ,
= (,,) ,
= (,,) .

Compute the coordinates of the fourth vertex
.

Perpendicular to two vectors

Let

,

be two space vectors. We have a vector
which is perpendicular to both
and
. What is this vector
?

Perpendicular and vector product

Let
be a space vector. Find the vector
who is perpendicular to
, such that the vector product u
is equal to (,,).

Linear relation

We have 4 space vectors:

,
,
,
.

Find 4 integers
,
,
,
such that

,

but the integers
,
,
,
are not all zero.

Scalar and vector products

Let
be a space vector. Find the vector
such that the scalar product and the vector product

,
.

Volume of parallelepiped

Compute the volume of the parallelepiped in the cartesian space having a vertex
, and such that the 3 vertices adjacent to
are

= (,,) ,
= (,,) ,
= (,,) .

Volume of tetrahedron

Compute the volume of the tegrahedron in the cartesian space whose 4 vertices are

= (,,) ,
= (,,) ,
= (,,) ,
= (,,) .

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