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Orthogonal vectors

Definition.
Two vectors a and b are orthogonal if they are perpendicular, i.e., angle between them is 90° (Fig. 1).
Orthogonal vectors
рис. 1
Condition of vectors orthogonality.Two vectors a and b are orthogonal, if their dot product is equal to zero.
a · b = 0

Examples of tasks

Examples of plane tasks

In the case of the plane problem for the vectors a = {ax; ay} and b = {bx; by} orthogonality condition can be written by the following formula:

a · b = ax · bx + ay · by = 0
Example 1. Prove that the vectors a = {1; 2} and b = {2; -1} are orthogonal.

Solution:

Calculate the dot product of these vectors:

a · b = 1 · 2 + 2 · (-1) = 2 - 2 = 0

Answer: since the dot product is zero, the vectors a and b are orthogonal.

Example 2. Are the vectors a = {3; -1} and b = {7; 5} orthogonal?

Solution:

Calculate the dot product of these vectors:

a · b = 3 · 7 + (-1) · 5 = 21 - 5 = 16

Answer: since the dot product is not zero, the vectors a and b are not orthogonal.

Example 3. Find the value of n where the vectors a = {2; 4} and b = {n; 1} are orthogonal.

Solution:

Calculate the dot product of these vectors:

a · b = 2 · n + 4 · 1 = 2n + 4
2n + 4 = 0
2n = -4
n = -2

Answer: vectors a and b are orthogonal when n = -2.

Examples of spatial tasks

In the case of the plane problem for the vectors a = {axayaz} and b = {bxbybz} orthogonality condition can be written by the following formula:

a · b = ax · bx + ay · by + az · bz = 0
Example 4. Prove that the vectors a = {1; 2; 0} и b = {2; -1; 10} is orthogonal.

Solution:

Calculate the dot product of these vectors:

a · b = 1 · 2 + 2 · (-1) + 0 · 10 = 2 - 2 + 0 = 0

Answer: since the dot product is zero, the vectors a and b are orthogonal.

Example 5. Are the vectors a = {2; 3; 1} and b = {3; 1; -9} orthogonal.

Solution:

Calculate the dot product of these vectors:

a · b = 2 · 3 + 3 · 1 + 1 · (-9) = 6 + 3 -9 = 0

Answer: since the dot product is zero, the vectors a and b are orthogonal.

Example 6. Find the value of n where the vectors a = {2; 4; 1} and b = {n; 1; -8} are orthogonal.

Solution:

Calculate the dot product of these vectors:

a · b = 2 · n + 4 · 1 + 1 · (-8)= 2n + 4 - 8 = 2n - 4
2n - 4 = 0
2n = 4
n = 2

Answer: vectors a and b are orthogonal when n = 2.

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