# 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 v1, 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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