Is warmth a scalar or a vector

Laws of calculation for vectors in coordinate representation

Addition and subtraction of vectors

Vectors are added or subtracted by adding or subtracting the corresponding components, in more detail here.

Example:

The three vectors are given:


Multiplication of a vector by a scalar

When multiplying a vector by a scalar, all components of the vector are multiplied by the scalar.

Example:

The three vectors are given:

Example:

The distance between two points P1 and P2 in three-dimensional space is to be determined.

The position vectors for the points are:

The amount of the connection vector of the two points corresponds to their distance from one another in three-dimensional space.

Note: When indexing the coordinates xij the first index stands for point P.i and the second index for the coordinate axis.


The scalar product

See also here.

The two vectors are given

We first formally multiply these with each other:

Note that for two mutually perpendicular vectors the scalar product is zero and the square of a unit vector is 1, the above expression is very simplified.

The scalar multiplication can also be carried out with column vectors:

example:

Which angle do both vectors include with each other?


The vector product

See also here.

Similar to the scalar product, one first multiplies formally. Then you simplify. To do this, you should look again at the rules for the cross product.
The following applies:

The following applies in particular to the basis vectors of the Cartesian coordinate system:

Formal multiplication:

The structure of the formula can be represented as a three-row determinant. This can be calculated according to Sarrus' rule, so that it provides the calculation formula. To apply Sarrus' rule, first write the first and second columns again after the determinant. Then all diagonal connections of three elements are formed, 3 times from top left to bottom right, and 3 times from bottom left to top right. Then products of these three factors are formed. The products created in this way are then summed up to a total in such a way that the products are counted as positive from top left to bottom right and negative from bottom left to top right.


Example:

The vectors are given

The vector product from both vectors is to be formed. The result must be checked with a suitable invoice.

The test can be carried out with the scalar product. The cross product of both vectors is a vector that is perpendicular to the plane spanned by the two vectors. As a result, the scalar product of the result vector with both vectors must result in the value zero.

Example:

The corner points A, B and C of a triangle have the coordinates
A (2 | -4 | 4); B (0 | 3 | 2) and C (-4 | -4 | 6).
Calculate the area of ​​the triangle. Then you should check the result with the formulas of plane trigonometry.
Preliminary consideration: A drawing should show the geometric representation.
The cross product of two vectors is a vector that runs perpendicular to the plane spanned by the two vectors and whose amount corresponds to the area of ​​the parallelogram that can be formed from the two vectors. The diagonals of the parallelogram divide it into two congruent triangles. This means that the triangular area is half the amount of the cross product.

The position vectors and the vectors of the triangle sides are determined:

The area of ​​the drawn parallelogram is obtained from the cross product:

Result control:

The sample confirms the first invoice.


Summary

Addition and subtraction of vectors

Multiplication of a vector by a scalar

The scalar product

The vector product

Here you will find an overview of all contributions and tasks on the subject of vector calculations.

You can buy these and other teaching materials in our shop. There teachers will find WORD files that they can change as required. You can also download all materials as PFD files for free.