Perpendicular vectors

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Vectors perpendicular in the plane are two vectors that form a 90 degree angle and their vector product is zero.

In other words, two vectors will be perpendicular when they form a right angle, and therefore their vector product will be zero.

To calculate whether one vector is perpendicular to another, we can use the formula for the dot product from the geometric point of view. That is, taking into account that the cosine of the angle they form will be zero. Therefore, to know which vector is perpendicular to another, we would only have to set the vector product equal to 0 and find the coordinates of the mysterious perpendicular vector.

Formula of two perpendicular vectors

The main idea of ​​the perpendicularity of two vectors is that their vector product is 0.

Given that given any 2 perpendicular vectors, their vector product will be:

Perpendicular vectors

The expression reads: "vector a is perpendicular to vector b".

We can express the above formula in coordinates:

Expression in coordinates

Graph of two perpendicular vectors

The previous vectors represented in a plane would have the following form:

Vectors perpendicular in the plane

Where we can extract the following information:

Perpendicularity of the vectors and the plane

The vector perpendicular to the plane is known as a normal vector and is indicated by an n, such that:

Normal vector

Demonstration

We can prove the condition that the product of two perpendicular vectors is zero in a few steps. Therefore, we only have to remember the formula of the vector product from the geometric point of view.

  1. Write the formula for the vector product from the geometric point of view:
Vector product

2. We know that two perpendicular vectors form an angle of 90 degrees. So, alpha = 90, such that:

Vector product with the cosine substituted

3. Next, we calculate the cosine of 90:

Angle cosine

4. We see that by multiplying the cosine of 90 with the product of the modules, everything is eliminated because they are multiplying by 0.

Vector product of two perpendicular vectors

5. Finally the condition will be:

Perpendicularity condition of two vectors

Example

Express the equation in terms of any vector that is perpendicular to the vector v.

To do this we define any vector p and leave its coordinates as unknowns since we know them.

Example

So, we apply the formula of the vector product:

Vector product

Finally, we express the vector product in coordinates:

Cross product in coordinates

We solve the previous equation:

Equation as a function of vector p

So, this would be the equation as a function of the vector p that would be perpendicular to the vector v.

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