Orthogonal vectors

In this publication, we will consider which vectors are called orthogonal, what condition must be met in this case. We will also analyze examples of solving problems on this topic.

Condition of orthogonality of vectors

Vectors a и b are orthogonal, if the angle between them is right (i.e. equal to 90°).

Note: The scalar product of orthogonal vectors is zero. This is the essential condition for their orthogonality.

a · b = 0

That is, if in the plane a = {a_x; to_y} и b = {b_x; b_y}then a · b = A_x · b_x + a_y · b_y = 0

Examples of tasks

Task 1

Let us prove that the vectors a = {2; 4} и b = {-2; 1} orthogonal.

Decision:

a · b = 2 · (-2) + 4 · 1 = 0

Therefore, the given vectors are orthogonal, since their scalar product is equal to zero.

Task 2

At what value n vectors a = {3; -9} и b = {6; n} orthogonal.

Decision:

a · b = 3 · 6 + (-9) · n = 0

18 – 9n = 0

n = 2

In this way, a и b are orthogonal for n equal to two.