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Vectors are numerical numbers that describe the actual amounts with a vector’s magnitude and direction. The length of the vector is defined as its magnitude and the path in which the vector points tell us about the direction. Directions of vectors can be represented in different structures. However, they are usually meant in degrees. The idea of the vector is 200 years old, but its use started in the late 19th century. The most preferred examples of vectors are speed and acceleration. In this upcoming article, we will learn about the characteristics of vectors, their formulas, and their uses, along with solved examples.
Scalar quantity can be any physical quantity that contains only the magnitude, and the direction of a vector is not given.
Examples of scalar quantity are:- length, time, mass, distance, volume, temperature, etc.
Define vector:
It is a Latin word that means “carrier.”
A vector is an item with magnitudes and the direction of a vector form. It is generally denoted by a symbol that shows the direction(→), and its length shows the size. The symbol which shows the vector has a sharpened stone, and its furthest edge is the tail. It is indicated as
→
V.
Examples of vector quantity are:- weight, momentum, force, Acceleration, velocity, displacement, etc.
What is the Magnitude and Direction of a Vector?
The magnitude of a Vector-
The magnitude of any physical quantity is defined as its length. For example, if the vector is a, the magnitude is represented as ∥a∥. In any case, speed, mass, distance, volume, temperature, and so on are scalar amounts. The scalar has only the magnitude, though the vectors have both magnitude and direction.
Operations in vector algebra:-
As in standard algebra, we do arithmetic operations like addition and subtraction. But in the case of multiplication, we have two vector methods like dot product and cross product.
Addition of Vectors:
If we consider there are two vectors, P and Q, then, at that point, the total of these two vectors can be performed when the tail of vector Q meets with the head of vector A. Furthermore, during this addition, the magnitude and direction of the vectors ought not to change. The vector addition keeps two significant formulas, which are;
Commutative Law: P + Q = Q + P
Associative Law: P + (Q + R) = (P + Q) + R
Subtraction Of Vectors:
Here, the direction of different vectors is turned backwards (reversed), and after that, the addition is performed on both the given vectors. Assume that P and Q are the vectors for which the subtraction strategy must be done, then, at that point, we modify the direction of another vector, say for Q, make it – Q. Presently, we want to add vector P and – Q. Accordingly, the directions of the vectors are inverse one another, yet the magnitude continues as before.
P – Q = P + (- Q)
Multiplication of a vector:-
Let us consider that K is a scalar quantity, and we multiply it by a vector A, then it will be a scalar multiplication represented by KA. If we consider that k is positive, then the direction of vector KA should be as same as that of vector A plus the magnitude of vector KA can be represented as |KA|.
Dot product:-
A scalar product is often called a dot product. It is denoted by a symbol dot (.) between two vectors. In this case, two equal-length coordinate vectors are multiplied to yield a single number. When we take the scalar product of two vectors, we get either a number or a scalar quantity. If P and Q are two vectors, the dot product of both vectors is given by;
P.Q = |P| |Q| cos P.Q = |P| |Q| cos θ = |P| |Q| cosθ
Where θ is the acute angle between P and Q
If P and Q are both pointing in the same direction, i.e., = 0°, then;
|P| |Q| P.Q = |P| |Q|
If both P and Q are orthogonal, i.e. = 90°, then;
P.Q = 0 [because cos 90° = 0]
If two vectors are given in vector algebra as;
[P1,P2,P3,P4,….,Pn] = [P1,P2,P3,P4,….,Pn] = [P1,P2,P3,P4, Q = [Q1,Q2Q3,Q4,….,Qn]
Then they are given their dot product by;
P.Q = P1Q1+P2Q2+P3Q3+………..PnQn Dot or Scalar Product
Cross product:-
The multiplication sign(x) between two vectors denotes a cross product. In a three-dimensional system, it is a binary vector operation. If P and Q are two independent vectors, the result of their cross product (P x Q) is perpendicular to both vectors and normal to the plane containing both vectors. It is symbolised by;
P x Q = |P| |Q| sin θ = |P| |Q| sinθ
Where θ is the acute angle between P and Q
The magnitude of vector algebra:
The magnitude of a vector is the length of the vector. The magnitude of the vector is denoted as ∥a∥.
Methods for the magnitude of vectors in two and three aspects, as far as their directions are, are determined on this page. For a two-layered vector a=(a1,a2), the equation for its magnitude is
∥a∥=√a12+a22−−−−−−.
For a three-layered vector a=(a1,a2,a3), the equation for its magnitude is
∥a∥=√a12+a22+a32−−−−−−−−−−.
The equation for the greatness of a vector can be summed up to inconsistent aspects. For example, a=(a1,a2,a3,a4) is a four-layered vector, the equation for its magnitude is
∥a∥=√a12+a22+a32+a42−−−−−−−−.
The direction of a Vector-
The heading point of a vector in the direction of the vector, or at least, the point it makes with the x-axis. A vector is drawn by a line with a bolt-on the top and an appropriate point at the opposite end. For example, velocity is a vector. It gives the magnitude at which the article moves alongside the direction the item is moving. The power vector also shows the direction in which power is applied. The direction of a vector is signified by →a=a^a
Where |a| signifies the magnitude of the vector, while ^a is a unit vector and signifies the bearing of the vector a.
Vector Algebra Example
Question: Find the dot product of the position vectors P(1, 3, – 4) and Q(3,−5, 2).
Ans: according to the vector variable based maths meaning of the spot item, we know;
P.Q = P1Q1+P2Q2+P3Q3+… .PnQn
Consequently,
P.Q = 1.3 + 3.(- 5) + (- 4).2
= 3 – 15 – 8
= -20
Conclusion
Vectors are geometrical objects with magnitude and direction. A vector has a beginning point and a terminal point, reflecting the point’s end location. Addition, subtraction, and multiplication are some of the mathematical operations performed on vectors. A unit vector is defined as a vector with a magnitude and direction of a vector. The magnitude of a vector formula determines the numeric value for a particular vector. A vector has both a magnitude and a direction of a vector. The magnitude of a vector formula is the total of the vector’s component measurements.
