Orthogonal Vectors – What is it, definition and concept

Two orthogonal vectors in the plane are two vectors that form a 90 degree angle and their dot product is zero.

In other words, two vectors are orthogonal if they form a right angle and therefore their dot product is zero.

The orthogonal word may seem difficult and quite technical at first, but once we understand the concept, it is easier to remember than it sounds.

Orthogonal and perpendicular

This word comes from ancient Greek and means right angle. If we think in geometric terms we can understand a right angle as two vectors that start from the same initial point. Defining the vertical vector as the vector to and the horizontal vector as the vector b, we can draw them as follows:

Orthogonal Vectors
Orthogonal Vectors

The vectors share the same starting point, so we can see that they form a right angle. So we can identify two orthogonal vectors if they form a right angle. We can also understand the meaning of orthogonal by thinking about perpendicular vectors.

If we repeat the previous representation in plane, it will look like this:

Orthogonal Vectors In The Plane
Orthogonal vectors in the plane

So we can determine if two vectors are orthogonal by plotting in the plane or by calculating the dot product. If the result of the scalar product is equal to zero, then the vectors are orthogonal.

In more advanced levels of geometry, a differentiation is made between orthogonal and perpendicular vectors depending on whether it is a Euclidean space or not. Since we are at a basic level of geometry, we can match orthogonality and perpendicularity. In this way, we can give it a touch of intellectual and speak of orthogonality instead of perpendicularity.

Characteristics

The main idea of ​​the perpendicularity of two vectors is that their dot product is 0. So, the multiplication of the coordinates of the vectors will be:

Scalar Product 2
Scalar product

The expression reads: “vector b is orthogonal to vector a”.

In terms of bases and linear combinations, if two vectors are orthogonal, so will their base.

Example

Show whether the following vectors are orthogonal vectors.

Scalar Product 3
Scalar product

Answer: the two previous vectors are not orthogonal since the dot product of both is different from zero. If the result of the dot product had been zero, then the two vectors would form an angle of ninety degrees.

For the dot product to be zero, it is enough that the coordinates of the vectors are complementary. In other words, if the first coordinate of the vector to is zero and the second coordinate of the vector b is zero, their dot product will be zero, and therefore they will form an angle of ninety degrees.

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