Properties of Vectors

Vectors are one of the most important concepts in mathematics. Vectors are quantities that have both magnitude and direction. A vector quantity is represented by an arrow above its head. Vectors help us understand the behaviour of directional quantities in 2D and 3D planes. Vectors are also used for determining the position and change of position of points.

Every vector follows a certain set of rules, known as the properties of vectors. It is highly important to know these properties to have a strong command of vector algebra. In this article, we will see the definition of a vector, the properties of vectors, and the properties of vector products.

Vectors are defined as,

A quantity that has both magnitude and direction.

Vectors are used to represent certain vector quantities like displacement, Force, velocity, electric field, Impulse , Angular momentum etc. Vectors are represent by a letter with an arrow over their head .

For example a vector [Tex]\overrightarrow{A}[/Tex] where A or |A| represents the magnitude of the vectors.

There are various basic properties of vectors are:

• Components of a Vector
• Magnitude of a Vector
• Direction of a Vector

Let’s discuss these in details as follows:

Components of a Vector

Vector in a coordinate system is often decomposed into its components along the axes of that system. In a Cartesian coordinate system, a vector is broken down into its x, y, and z components (in three-dimensional space). These components describe how much of the vector lies along each axis.

In a 3D Cartesian coordinate system, a vector v can be represented as:

[Tex] \mathbf{v} = \langle v_x, v_y, v_z \rangle [/Tex]

Where v_x, v_y and v_z are the components of the vector along the x-axis, y-axis and z-axis respectively.

In three dimensions, if the angles with respect to the x, y, and z axes are α, β, and ℽ respectively

• [Tex] v_x = | \mathbf{v} | \cos \alpha [/Tex]
• [Tex] v_y = | \mathbf{v} | \cos \beta [/Tex]
• [Tex] v_z = | \mathbf{v} | \cos \gamma [/Tex]

Read More about Components of a Vector.

Magnitude of a Vector

Magnitude of a vector, often referred to as its length or norm, is a measure of how long the vector is. It is a scalar quantity that represents the distance from the vector’s initial point to its terminal point in the coordinate system.

The magnitude of a vector v, denoted as ∣v∣, is calculated using the components of the vector.

For a vector [Tex]\mathbf{v} = \langle v_x, v_y \rangle[/Tex] in a 2D Cartesian coordinate system, the magnitude is given by:

[Tex]| \mathbf{v} | = \sqrt{v_x^2 + v_y^2} [/Tex]

For a vector [Tex] \mathbf{v} = \langle v_1, v_2, \ldots, v_n \rangle[/Tex] in an n-dimensional space, the magnitude is given by:

[Tex]| \mathbf{v} | = \sqrt{v_1^2 + v_2^2 + \cdots + v_n^2} [/Tex]

Direction of a Vector

Direction of a vector specifies the orientation of the vector in space. It indicates where the vector is pointing relative to a reference direction, usually the positive x-axis in a Cartesian coordinate system.

For a vector [Tex]\mathbf{v} = \langle v_x, v_y, v_z \rangle [/Tex] in 3D space, the direction is often described using two angles:

The angle α, β, and ℽ with the x, y and z-axis respectivly.

Vector operations are operations defined on vectors to get sum results:

• Addition of Vectors
• Subtraction of Vectors
• Scalar Multiplication
• Equality of Vectors

Let’s discuss these operations in detail:

Addition of Vectors

Vector Addition is a fundamental operation in vector mathematics that combines two or more vectors to produce a resultant vector. This operation is essential in physics, engineering, computer science, and many other fields.

Vectors can be added using various methods such as

• Graphical Method
• Triangle Law
• Parallelogram Law
• Polygon Law

Properties of Vector Addition

Some of the common properties of vector addition are:

Commutative Property: Vector addition is commutative. For example , for two vector a and b this property say a +b = b + a.

Associative property: Vector addition is Associative in nature. For example , for three vector a , b and c this property say (a +b)+c = a+(b + c ).

Distributive property: Vector addition is distributive over Scaler Multiplication. For example λ(a + b ) = λa + λb , where λ is a scalar quantity and a and b are two vectors .

Existence of Inverse For every vector a there exists a another vector b such that a + b =0 i.e. b = -a .

Existence of Identity : For every vector a there exists a another vector b such that a + b = a i.e. b = 0.

Note: It’s important to recognize that two vectors can be added only if they are of same nature.

Subtraction of Vectors

Two vectors can be subtracted to get a new resultant vector. Vector subtraction is just similar to vector addition.

Vector Subtraction can be seen as addition of first vector and the negative or second vector or we can say that for two vectors a and b , a – b = a + (-b). Vector subtraction do follows the laws of vector addition just like vector addition.

Properties of Vector Subtraction

Some of the common properties of vector subtraction are:

• Commutative Property: Vector subtraction isn’t commutative i.e. a – b ≠ b – a.
• Associative Property: Vector subtraction is Associative in nature. For example , for three vector a , b and c this property say (a – b) – c = a- ( b – c ).
• Distributive Property: Vector subtraction is distributive over Scaler Multiplication. For example λ(a – b ) = λa – λb , where λ is a scalar quantity and a and b are two vectors.
• Existence of Inverse: For every vector a there exists a another vector b such that a – b =0 i.e. b = a.
• Existence of Identity: For every vector a there exists a another vector b such that a – b = a i.e. b = 0.

Scalar Multiplication

Scalar multiplication is an operation in vector mathematics where a vector is multiplied by a scalar (a real number). This operation scales the magnitude of the vector without changing its direction (unless the scalar is negative, which also reverses the direction).

For a vector [Tex]\mathbf{v} = \langle v_x, v_y, v_z \rangle[/Tex] and a scalar k:

[Tex]k\mathbf{v} = k \langle v_x, v_y, v_z \rangle = \langle k v_x, k v_y, k v_z \rangle[/Tex]

Equality of Vector

Two vector are said to be equal if they have the same magnitude and direction.

This means that if you have two vectors, V and W then they are equal only if ∣v∣=∣w∣ and they point in the same direction. In other words, they represent the same physical quantity or geometric quantity in space. For example two vector X = ai + bj + ck , and Y = pi + qj + rj are equal iff a = p , b= q and c =r.

Other then all the discussed properties, some other properties or operations on vectors are:

• Scalar Product or Dot Product
• Vector Product or Cross Product

Let’s discuss these in detail.

Dot Product

Dot Product, also known as the scalar product, is an algebraic operation that takes two vectors and returns a single scalar quantity. It is a measure of the extent to which two vectors are parallel and can be used to find the angle between the vectors.

For two vectors \(\mathbf{a}\) and \(\mathbf{b}\), the dot product is defined as:

[Tex]\mathbf{a} \cdot \mathbf{b} = \sum_{i=1}^n a_i b_i [/Tex]= |a||b| cos θ

Where,

• a_i and b_i are the components of vectors,
• a and b, respectively,
• n is the number of dimensions, and
• θ is the angle between both vectors.

Properties of Scalar Product of Vectors

Commutative Property: The dot product of two vectors is commutative i.e. a.b = b.a or |a| * b|cosθ = |b| * |a|cosθ.

Associative Property: The dot product is not associative in Nature or we can say that (a.b).c ≠ a.(b.c) , it is because (a.b).c represent a scalar (a.b ) multiplied to c vector whereas a.(b.c) represent a scalar (b.c) multiplied to a vector.

Distributive Property: The dot product of vectors is distributive over vector addition. So we can say that for three vector a , b and c

a.(b+c) = a.b + a.c

Orthogonality: Scalar product of two perpendicular vectors is 0 .Two vectors whose dot product is zero are known as orthogonal vectors.

Scalar product of two Parallel/Collinear/Like Vectors is the product of magnitude of the two vectors. Also the dot product of a vector with itself is the square of its magnitude i.e. a. a = |a|².

Cross Product

Cross product is the vector product of vectors. It represent the product of magnitude of the vectors and the sine of angle between them.

Unlike the scalar product give results a scalar value , Vector product always give a vector which is perpendicular to both the given vector. The vector product of two vectors a and b with an angle α between them is mathematically calculated as

a × b = |a| |b| sin α n̂

Where,

• ∣a∣ and ∣b∣ are the magnitudes of vectors a and b,
• α is the angle between the vectors, and
• n̂ is the unit vector in the direction of a × b.

Note: The direction of the resultant vector is determined by the right-hand rule.

Properties of Vector Product of Vectors

Commutative Property: The Cross product of two vectors isn’t commutative i.e. a × b ≠ b × a or |a| * b|sinθ ≠ |b| * |a|sinθ.

Associative property: The Vector product is Not associative in Nature or we can say that (a × b)×c ≠ a × (b × c) .

Distributive Property: The Cross product of vectors is distributive over vector addition. So we can say that for three vector a , b and c

a × (b+c) = a × b + a × c

Collinearity: Vector product of two Parallel / collinear /like vectors is 0 .

Vector product of two perpendicular vector is equal to product of the magnitude of the two vectors. Also the Vector product of a vector with itself is the square of its magnitude ie. a × a = |a|².

Some of the common applications of vectors are:

• Physics
  • Representing forces, velocities, and accelerations.
  • Calculating work done by a force.
  • Describing electric and magnetic fields.
• Engineering
  • Analyzing forces and moments in structures.
  • Representing displacements and velocities in mechanics.
• Computer Graphics
  • Modeling and transforming objects.
  • Calculating lighting and shading effects.
  • Determining normals to surfaces.
• Navigation
  • Determining direction and speed of movement.
  • Pathfinding and route optimization.
• Robotics
  • Controlling the movement and orientation of robotic arms.
  • Path planning and navigation.
• Machine Learning and Data Science
  • Representing data points in high-dimensional space.
  • Calculating distances and similarities between data points.

Read More about the scalar and vector product of vector.

Question 1 : Find the value of λ if λ (a + b ) = c where a = 3i + 4j + k , b = i – j + 4k and c = 8i + 6j + 10k ?

Solution:

Given ,

a = 3i + 4j + k , b = i – j + 4k and c = 8i + 6j + 10k

Now λ (a + b ) = c can be written as ,

λa + λb = c

or

λ(3i + 4j + k ) + λ ( i – j + 4k) = 8i + 6j + 10k

comparing coefficients of i / j / k

4λ = 8 or

λ =2.

Question 2 : Find the additive Inverse of the vector A = 13i + 12j + 5k ?

Solution:

The additive inverse of a given vector is the negative of the same vector. So additive inverse of A is -13i – 12j – 5k

Question 3 : Calculate the work done by a Force of F = 2i – j + 4k , N that creates a displacement of D = 6i – 2j +k.

Solution:

Work done by force is given by the dot product of Force and displacement.

So work done W is ,

⇒ W = F.D = ( 2i – j + 4k ). (6i – 2j +k )

⇒ W = 12 + 2 + 4 = 18 J

So Work done by the given force equals to 18 Joules.

Question 4: If a and b are two vectors such that |a| = 2, |b| = 1/√2, and Find the angle between a and b , such that their cross product is a unit vector ?

Solution:

Given , cross product of a and b is a unit vector , |a × b| = 1.

Also ,

sinθ = |a × b| / (|a| |b|)

⇒ sinθ = 1/ (2 × 1/√2)

⇒ sinθ= 1/√2

⇒ θ= 45° , So angle between a and b is 45° .

How to calculate difference of two vectors ?

Difference of two vectors is similar to addition and follows the same laws. It can be seen as the addition of the first vector and the negative of the second vector.

How to calculate Area of parallelogram from vector product ?

The Area parallelogram can be found by calculating the magnitude of Vector product of the two given vectors.

How is the direction of cross product determined ?

Direction of the vector product can be determined from the right hand thumb rule , where the thumb points in the direction of the resultant vector when the fingers curl from the first vector to the second vector.

Can vectors with different units be added or subtracted ?

No , vectors with different units can’t be added or subtracted. They must have the same units or dimensions.

Why the dot product of orthogonal vector zero ?

Angle between two orthogonal vectors is 90° and cos90° is zero , so dot product of orthogonal vectors is zero.