Orthogonal vectors are vectors that are perpendicular to each other and form an angle of 90∘. In vector algebra, two vectors are said to be orthogonal when their dot product is zero. Orthogonality is the property that describes the perpendicular relationship between two vectors.
Vectors a and b are orthogonal vectors if
\vec{a}⋅\vec{b} = 0
The dot product of two n-dimensional vectors is given by
\vec{a}⋅\vec{b} = a_{1}b_{1} + a_{2}b_{2} + a_{3}b_{3} +⋯+ a_{n}b_{n}
Properties
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 non-zero orthogonal vectors has maximum magnitude.
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.
Consider two vectors a and b, the orthogonal projection of "b" onto "a", is expressed as:
(
(proj)_{\vec{a}}(\vec{b}) = \frac{a ⋅ b}{(|a|²)}\times\vec{a}
where,
- a ⋅ b is Dot Product
- |a| denotes Magnitude of a
This projection helps in understanding how much of vector "b" lies in the direction of vector "a".
Dot Product of Orthogonal Vectors
The dot product of two vectors is
\vec{a}⋅\vec{b} = a_{1}b_{1} + a_{2}b_{2} + a_{3}b_{3} +⋯+ a_{n}b_{n}=0
This property is the most common method used to determine whether two vectors are perpendicular.
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 vectors a and b is:
The magnitude of the cross product is:
|\vec{a} × \vec{b}| = |\vec{a}||\vec{b}|sin(θ) ,where θ is the angle between a and b
For orthogonal vectors, θ =90 degree therefore,
|\vec{a} × \vec{b}| = |\vec{a}||\vec{b}|
Solved Examples
1. For vector a = 3i + 4j and b = i + 2j, find projection of a on b.
Given, a = 3i + 4j and b = i + 2j
a.b = (3i + 4j).(i + 2j) = 3 + 8 = 11
b.b = 12 + 22 = 1 + 4 = 5
Projection of b on a,
Projb(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 and 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 and 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
Practice Problems
1. Determine whether the vectors
2. Determine whether the vectors
3. Find the value of x such that the vectors
4. For the vectors
- Find
\vec{a}.\vec{b} - Determine whether the vectors are orthogonal.
- Find the projection of a onto b.
5. Consider the vectors u = (1,1,0),v = (1,−1,0), w = (0,0,2) .Verify whether the vectors are pairwise orthogonal.