Orthogonal Vectors: Definition, Formula and Examples

Orthogonal vectors are a fundamental concept in linear algebra and geometry. Orthogonal vectors are vectors that are perpendicular to each other, meaning they meet at a right angle (90 degrees). Two vectors are orthogonal if their dot product is zero. In this article, we will learn about, Orthogonal Vectors Definition, Orthogonal Vectors Formula, Orthogonal Vectors Examples and others in detail.

Orthogonal Vectors Definition

Vectors a and b are orthogonal vectors if:

\vec{a}⋅\vec{b} = 0

Here, the dot product a⋅b is calculated as:

\vec{a}⋅\vec{b} = a_{1}b_{1} + a_{2}b_{2} + a_{3}b_{3} +⋯+ a_{n}b_{n}

What are Orthogonality of Vectors?

Orthogonality in vectors is a concept in mathematics where two vectors are said to be orthogonal vectors if they are perpendicular to each other. This is a key concept in vector algebra and has significant applications in various fields such as physics, engineering, and computer science.

Definition of Orthogonality

Two vectors are orthogonal vectors if their dot product is zero. Geometrically, this means the vectors form a right angle (90°) with each other. Mathematically, for vectors "a" and "b", orthogonality is defined as: "a ⋅ b" equals zero.

Orthogonal Vector Formula

To check the orthogonality of two vectors "a" and "b", represented as "\vec{a} = a_{1}i + a_{2}j + a_{3}k" and "\vec{b} = b_{1}i + b_{2}j + b_{3}k" ", we use their dot product: And Orthogonal Vector Formula is:

\vec{a} ⋅ \vec{b} = a_{1}b_{1} + a_{2}b_{2} + a_{3}b_{3} = 0

Properties of Orthogonal Vectors

Various properties of Orthogonal vectors are:

• Perpendicular Nature: Orthogonal vectors are always perpendicular to each other.

• Dot Product: The dot product of orthogonal vectors is zero.

• Null Vector: A null vector (a vector with zero magnitude) is orthogonal to every vector.

• Cross Product: The cross product of two orthogonal vectors is not zero unless one of the vectors is a null vector.

Orthogonal Projection of a Vector

Orthogonal projection of a vector is a concept in linear algebra where one vector is projected onto another vector in a perpendicular manner. This projection results in a new vector that lies along the direction of the vector onto which the projection is made.

Given two vectors "a" and "b", the orthogonal projection of "b" onto "a", denoted as (proj)_{b}(a), is calculated as the scalar projection of "b" onto "a", divided by the magnitude of "a", and then multiplied by "a". Mathematically, it can be expressed as:

((proj)_{b}(a) = (a ⋅ b) / (|a|²) × a

where:

• "a ⋅ b" represents Dot Product of Vectors "a" and "b",
• "|a|²" denotes Squared Magnitude of vector "a".

This projection helps in understanding how much of vector "b" lies in the direction of vector "a", thereby facilitating various calculations and analyses in vector algebra and geometry.

How to Check if Two Vectors are Orthogonal?

If the dot product of vectors"\vec{a}" and "\vec{b}" equals zero, then vectors "\vec{a}" and "\vec{b}" are orthogonal. If the dot product is not zero, then vectors "\vec{a}" and "\vec{b}" are not orthogonal.

Dot Product of Orthogonal Vectors

For two vectors a and b, the dot product is calculated as:

Dot product a · b equals the sum of the products of their corresponding components:

\vec{a}⋅\vec{b} = a_{1}\times\ b_{1} + a_{2}\times\ b_{2} + a_{3}\times\ b_{3} +⋯+ a_{n}\times\ b_{n}

If the vectors are orthogonal, meaning they're perpendicular, then their dot product is zero:

\vec{a}⋅\vec{b} = 0

This property is crucial in various mathematical and practical applications, particularly in linear algebra and geometry.

Cross Product of Orthogonal Vectors

When two vectors are orthogonal (or perpendicular), their cross product results in a new vector that is perpendicular to both of the original vectors.

The cross product of two vectors \vec{a} and \vec{b}, denoted by \vec{a}\times \vec{b}, produces a new vector that is perpendicular to both a and b. This new vector's direction follows the right-hand rule: if you align your right hand's fingers along vector a and then curl them toward vector b, your thumb will point in the direction of the resulting cross product vector.

Magnitude of the cross product is determined by the magnitudes of the original vectors and the sine of the angle between them. If a and b are orthogonal, meaning the angle between them is 90 degrees, then the sine of that angle is 1. Therefore, the magnitude of the cross product is simply the product of the magnitudes of a and b.

The formula for the magnitude of the cross product of a and b is:

Cross Product of a and b = \vec{a} × \vec{b} = |a| × |b| × sin(θ)

where,

• θ is the angle between a and b

However, if a and b are not orthogonal, the magnitude of their cross product is reduced by the angle between them.

Conclusion

In conclusion, orthogonality of vectors is a fundamental concept in mathematics and has wide-ranging applications in various fields. When the dot product of two vectors is zero, they are called orthogonal vectors, indicating perpendicularity. This property is utilized in numerous calculations and analyses across disciplines like physics, engineering, and computer science.

Related Article on Orthogonal Vectors:

• Scalar and Vector
• Vector Calculus
• Vector Operations
• Triangle Law of Vector Addition
• Parallelogram Law of Vector Addition

Examples questions on Orthogonal Vectors

1: For vector a = 3i + 4j and b = i + 2j, find projection of a on b.

Given,

• a = 3i + 4j
• b = i + 2j

a.b = (3i + 4j).(i + 2j) = 3 + 8 = 11

b.b = 1² + 2² = 1 + 4 = 5

Projection of b on a,

Proj_b(a) = 11/5(i + 2j)

2: For vector a = i and b = j, find dot a on b and check whether they are orthogonal vectors or not.

Given,

• a = i + 0j
• b = 0i + j

Dot product of a and b: (1)(0) + (0)(1)

= 0 + 0 = 0

Thus, 'a' and 'b' are orthogonal.

3: For vector a = 2i + 2j and b = 2i - 2j, find dot a on b and check whether they are orthogonal vectors or not.

Given,

• a = 2i + 2j
• b = 2i - 2j

Dot product of a and b: (2)(-2) + (2)(2)

= -4 + 4 = 0

Thus, 'a' and 'b' are orthogonal.

4: For vector a = i + 0j + 0k and b = 0i + 1j + 0k, and c = 0i + 0j + 1k, check whether they are pair wise orthogonal vector or not.

Given,

• a = i + 0j + 0k
• b = 0i + 1j + 0k
• c = 0i + 0j + 1k

Dot product of a and b: (1)(0) + (0)(0) + (0)(0)

= 0

Dot product of a and b: (0)(0) + (1)(0) + (0)(0)

= 0

Dot product of a and b: (0)(0) + (0)(0) + (1)(0)

= 0

Thus, 'a' and 'b' , 'b' and 'c', and 'c' and 'd' are pair wise orthogonal.