
Orthogonal vectorsPage Navigation:
Definition. Two vectors a and b are orthogonal if they are perpendicular, i.e., angle between them is 90° (Fig. 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 tasksExamples of plane tasksIn the case of the plane problem for the vectors a = {a_{x}; a_{y}} and b = {b_{x}; b_{y}} orthogonality condition can be written by the following formula: a · b = a_{x} · b_{x} + a_{y} · b_{y} = 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 = 0Answer: 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 = 16Answer: 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 + 42n + 4 = 0 2n = 4 n = 2 Answer: vectors a and b are orthogonal when n = 2. Examples of spatial tasksIn the case of the plane problem for the vectors a = {a_{x}; a_{y}; a_{z}} and b = {b_{x}; b_{y}; b_{z}} orthogonality condition can be written by the following formula: a · b = a_{x} · b_{x} + a_{y} · b_{y} + a_{z} · b_{z} = 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 = 0Answer: 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 = 0Answer: 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  42n  4 = 0 2n = 4 n = 2 Answer: vectors a and b are orthogonal when n = 2.
Vectors
Vectors Definition. Main information
Component form of a vector with initial point and terminal point
Length of a vector
Direction cosines of a vector
Equal vectors
Orthogonal vectors
Collinear vectors
Coplanar vectors
Angle between two vectors
Vector projection
Addition and subtraction of vectors
Scalarvector multiplication
Dot product of two vectors
Cross product of two vectors (vector product)
Scalar triple product (mixed product)
Linearly dependent and linearly independent vectors
Decomposition of the vector in the basis
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