Vector Form

Vector Form is used to represent a point or a line in a cartesian system, in the form of a vector. The vector form of representation helps to perform numerous operations such as addition, subtractions, multiplication of vectors. The cartesian form of representation of a point (x, y, z) can be written in vector form as \(\vec A = x\hat i + y\hat j + z\hat k\).

Let us understand the use of vector form to represent a point, a line, a plane, with the help of examples, FAQs.

Vector form helps to represent any point in space as a vector. A point P in space is represented as a vector \(\rightarrow OP\), which represents the vector line OP connecting the line from the origin O to the point P. The vector form is used to represent a line or a plane in a three-dimensional space. The point P(x, y, z) can be represented as a vector line \(\vec OP = x\vec i + y\vec j + z\vec k\).

The vector form is useful to perform numerous arithmetic operations involving vectors. Vector form is useful to simultaneously represent the magnitude and direction of any physical quantity. The forms of vectors are also called Euclidean vectors or Spatial vectors. Vectors have many applications in maths, physics, engineering, and various other fields.

Representation of Vectors

Vectors are usually represented in bold lowercase such as a or using an arrow over the letter as \(\vec{a}\). Vectors can also be denoted by their initial and terminal points with an arrow above them, for example, vector AB can be denoted as \(\overrightarrow{AB}\). The standard form of representation of a vectors is \(\vec{A}=a \hat{i}+b\hat{j}+c\hat{k}\). Here, a,b,c are real numbers and \(\hat{i}, \hat{j}, \hat{k}\) are the unit vectors along the x-axis, y-axis, and z-axis respectively.

The initial point of a vector is also called the tail whereas the terminal point is called the head. Vectors form is used to describe the movement of an object from one place to another. In the cartesian system, vectors can be denoted as points in a coordinate system. Similarly, vectors in 'n' dimensions can be denoted by an 'n' tuple.

Let us now check a few of the important concepts related to vectors.

Direction Ratios

The point A,(a, b, c) in a three-dimensional cartesian system is represented in vector form with the position vector as \(\vec OA = a\vec i + b\vec j + c\vec k \) and has the direction ratios a, b, c. This ratio represents the vector line with reference to the x-axis, y-axis, and z-axis respectively. Further, these direction ratios also help to derive the direction cosines.

Direction Cosine

Direction Cosine gives the relation of a vector or a line in a three-dimensional space, with each of the three axes. The direction cosine is the cosine of the angle subtended by this line with the x-axis, y-axis, and z-axis respectively. If the angles subtended by the line with the three axes are α, β, and γ, then the direction cosines are Cosα, Cosβ, Cosγ respectively. The direction cosines for a vector \(\overrightarrow A = a \hat i + b \hat j + c \ \hat k\) is Cosα = \(\frac{a}{\sqrt {a^2 + b^2 + c^2}}\), Cosβ = \(\frac{b}{\sqrt {a^2 + b^2 + c^2}}\), Cosγ = \(\frac{c}{\sqrt {a^2 + b^2 + c^2}}\). The direction cosines are also represented by l, m, n, and we can prove that l² + m² + n² = 1.

Magnitude of Vectors

The magnitude of a vector can be calculated by taking the square root of the sum of the squares of its components. If (x,y,z) are the components of a vector A, then the magnitude formula of A is given as follows. The magnitude of a vector is a scalar value

|A| = √ (x²+y²+z²)

Angle Between Two Vectors

The angle between two vectors can be calculated using the dot product formula. Let us consider two vectors a and b and the angle between them to be θ. Then, the dot product of two vectors is given by a·b = |a||b| cosθ. We need to determine the value of the angle θ. The angle between two vectors also indicates the directions of the two vectors. θ can be evaluated using the following formula:

θ = cos^-1[(a·b)/|a||b|]

The vector form of the equation of a line uses the point or a line required to form the equation of a line in vector form. The two ways of forming a vector form of equation of a line is as follows.

• The vector form of the equation of a line passing through a point having a position vector \(\vec a\), and parallel to a vector line \(\vec b\) is \(\vec r = \vec a + λ\vec b\).
• The vector form of the equation of a line passing through two points with the position vector \(\vec a\), and \(\vec b\) is \(\vec r = \vec a + λ(\vec b - \vec a)\).

The vector form of the equation of a plane in cartesian coordinate system can be computed through different methods, based on the available inputs values about the plane. The following are the four different expressions for the equation of a plane in vector form.

• Normal Form: Equation of a plane at a perpendicular distance d from the origin and having a unit normal vector \(\hat n \) is \(\overrightarrow r. \hat n\) = d.
• Perpendicular to a given Line and through a Point: The equation of a plane perpendicular to a given vector \(\overrightarrow N \), and passing through a point \(\overrightarrow a\) is \((\overrightarrow r - \overrightarrow a). \overrightarrow N = 0\)
• Through three Non Collinear Lines: The equation of a plane passing through three non collinear points \(\overrightarrow a\), \(\overrightarrow b\), and \(\overrightarrow c\), is \((\overrightarrow r - \overrightarrow a)[(\overrightarrow b - \overrightarrow a) × (\overrightarrow c - \overrightarrow a)] = 0\).
• Intersection of Two Planes: The equation of a plane passing through the intersection of two planes \(\overrightarrow r .\hat n_1 = d_1\), and \(\overrightarrow r.\hat n_2 = d_2 \), is \(\overrightarrow r(\overrightarrow n_1 + λ \overrightarrow n_2) = d_1 + λd_2\).

Related Topics

The following topics help in a better understanding of vector form.

• Zero Vector
• Vector Algebra
• Multiplication Of Vectors
• Types Of Vectors
• Projection Vector

What Is Vector Form?

Vector form helps to represent any point in space as a vector. A point P in space is represented as a vector \(\rightarrow OP\), which represents the vector line OP connecting the line from the origin O to the point P. The vector form is used to represent a line or a plane in a three-dimensional space. The point P(x, y, z) in a cartesian plane can be represented as a vector line \(\vec OP = x\vec i + y\vec j + z\vec k\).

How Do We Convert A Cartesian Form To Vector Form?

The cartesian form of representation of a point A(x, y, z), can be easily written in vector form as \(\vec A = x\hat i + y\hat j + z\hat k\). Here \(\hat i\), \(\hat j\), and \(\hat k\) are the unit vectors along the x-axis, y-axis, and z-axis respectively.

How Do We Represent A Point In Vector Form?

A point P having the coordinates (x, y, z) is represented in vector form as a position vector \(\overrightarrow {OP}\), where O is the origin and P is the required point and the vector form of representation of the point P is \(\vec P = x\hat i + y\hat j + z\hat k\).

How Do We Represent A Line In Vector Form?

The vector form of the equation of a line uses the point or a line required to form the equation of a line in vector form. There are two ways of representing a line in vector form.

• The vector form of the equation of a line passing through a point \(\vec a\), and parallel to a vector line \(\vec b\) is \(\vec r = \vec a + λ\vec b\).
• The vector form of the equation of a line passing through two points \(\vec a\), and \(\vec b\) is \(\vec r = \vec a + λ(\vec b - \vec a)\).

How Do We Represent A Plane In Vector Form?

• Equation of a plane at a perpendicular distance d from the origin and having a unit normal vector \(\hat n \) is \(\overrightarrow r. \hat n\) = d.
• The equation of a plane perpendicular to a given vector \(\overrightarrow N \), and passing through a point \(\overrightarrow a\) is \((\overrightarrow r - \overrightarrow a). \overrightarrow N = 0\)
• The equation of a plane passing through three non collinear points \(\overrightarrow a\), \(\overrightarrow b\), and \(\overrightarrow c\), is \((\overrightarrow r - \overrightarrow a)[(\overrightarrow b - \overrightarrow a) × (\overrightarrow c - \overrightarrow a)] = 0\).
• The equation of a plane passing through the intersection of two planes \(\overrightarrow r .\hat n_1 = d_1\), and \(\overrightarrow r.\hat n_2 = d_2 \), is \(\overrightarrow r(\overrightarrow n_1 + λ \overrightarrow n_2) = d_1 + λd_2\).

What Are The Uses Of Vector Form?

The vector form is useful to perform numerous arithmetic operations on vectors. The arithmetic operation of addition, subtraction, dot product, the cross product of vectors, and the projection of vectors, can be easily performed with vector form of representation.