#
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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- Description: collection of exercises on 3D vectors. This is the main site of WIMS (WWW Interactive Multipurpose Server): interactive exercises, online calculators and plotters, mathematical recreation and games
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