Definition of Vector and Scalar Linear Algebra

In this article we will discuss the meaning of vectors, especially for vectors on R² and R³. Many quantities in physics such as force, speed, acceleration, displacement, and shift are vectors that can be expressed as directional line segments. The algebraic view, examines the properties of algebra from a vector space, that is, the properties of vector addition and scalar vector multiplication. In this article the properties will be summarized in the discussion on R² and R³. In addition to physics, vector understanding is also widely used in other fields outside mathematics such as technology, economics, biology and so on.

Vector and Vector Operation

A solution of a system with m linear equations in n unknown numbers is a tuple-n of real numbers. We will call a tuple-n real numbers as a vector. If an n-tuple is expressed as a matrix of 1 x n, we will call it a row vector. Conversely, if the tuple is expressed by a matrix n x 1, we call it a column vector. For example the solution of the linear system below,

x₁ + x₂ = 5

x₁ – x₂ = 3

can be expressed by row vectors (2, 1) or column vectors 

then in this section we will explain the presentation and understanding of vectors in geometry and algebra.

Vector Definition of Geometry

Many quantities that we encounter in science such as: length, mass, volume, and electric charge, can be expressed by a number.

Such magnitude (and the number that becomes its size) is called Scalar, There are other quantities, such as speed, force, torque, and shifts to describe it require not only numbers, but also directions. Such magnitude is called a Vector. In this article vectors in space 2 and space 3 will be introduced geometrically, and we will discuss some of the basic characteristics of this operation.

Vectors can be expressed geometrically as directed line segments or arrows in space 2 or space 3. The length of the arrow is the magnitude of the vector while the direction of the arrow is the direction of the vector. The arrow has a base and a tip (Figure 1). The base of the arrow is called the initial point and the tip of the arrow is called the terminal point.

We will paint a vector with bold letters, for example u and v, because this is rather difficult to do in writing, you can describe vectors with symbols 

When discussing vectors, we will declare numbers as scalars. All scalars are real items and will be expressed by ordinary lowercase letters, for example a, b, c, and k.

If, as in figure 1, the starting point of the vector v is A and the terminal point is B, we write it down.

Vectors that have the same length and direction, such as the vectors in Figure 2, are called equivalents. Because we want a vector that is determined by its length and direction, the equivalent vectors are considered equal even though the vectors may be placed in different positions.

Vector operations

For two or more vectors, operations can be carried out as follows,

1. Vector addition and subtraction.
2. Vector multiplication with scalar.

Provisions in Addition and Reduction of Vector

To obtain the resultant of two vectors u and v, move v without changing the size and direction until the base coincides with the tip u. Then u + v is a vector that connects between the base u and the end v. This method is called the triangular law, which is illustrated in figure 3. Another way of describing u + v is to move v so that the base coincides with the base u. Then u + v is a vector which denies u and which coincides with the diagonal parallelogram whose sides are u and v. This method is called the parallelogram law, which is illustrated in figure 4.

You can prove yourself that this sum is commutative, namely: u + v = v + u.

Vector addition properties

Commutative properties: u + v = v + u.

Associative properties: (u + v) + w = u + (v + w)

The sum of several vectors does not need to depend on the sequences. Addition can be expanded as shown in figure 6, i.e.,

u = u₁ + u₂ + u₃ + u₄ + u₅

This method is called the polygon method

Vector Multiplication Provisions with Scalar

If u is a vector, then 3u is a vector that is in the direction of u but whose length is three times the length u, the vectors are -2u twice the length u but in the opposite direction (Figure 7). In general, cu is a scalar multiple of the u vector, whose length is | c | times the length u, in the direction of u if c is positive and in the opposite direction if c is negative. Specifically, (-1) u (also written as -u) is the same length as u, but the direction is opposite. This vector is called a negative vector u because if - u is summed with u, the result is a zero vector (i.e. a point), this vector is the only vector without a certain direction, called the zero vector, which is denoted by 0. This vector is the sum element u + 0 = 0 + u = u. Finally, the reduction is determined as

u – v = u + (-v)

Example 1.

In figure 8, it is expressed w with u and v

Solution:

Because u + w = ​​v, then

w = v- u

Example 2. 

In figure 9,   specify that m is in u and v.

In general, if  with 0 < t < 1, then

m = (1-t)u + tv 

The evidence we get for m can also be written as

u + t(v - u)

If t changes from - ∞ to + ∞ we get all the vectors leading to the lines shown in the following figure,

Vector Definition of Algebra

Problems involving vectors can often be simplified by introducing a Cartesian coordinate system. Following this we will limit the discussion of vectors in space-2 and space-3.

Cartesian coordinates in space-2

We begin by taking a Cartesian coordinate system in the plane. As a representative of u vector, we select an arrow that starts at the origin (Figure 10). This arrow is determined singly by the coordinates u₁ and u₂ end points. This means that the vector u is determined by an ordered pair (u₁, u₂) by introducing the Cartesian coordinate system, which is illustrated in Figure 11. So we assume (u₁, u₂) is a vector u. This ordered pair (u₁, u₂) is a vector u algebraically.

To form such a coordinate system, we select an arrow that starts at point 0 as the origin and select two lines that are perpendicular as the coordinate axis through that origin, which is illustrated in Figure 10 below.

Mark these axes with x and y, then choose a positive direction for each coordinate axis and also a unit of length to measure distance.

Then we determine the arrow point by the coordinates u₁ and u₂.

This means that the vector u is determined by an ordered pair (u₁, u₂).

Vector Operations

Two vectors u =  (u₁, u₂)  and v = (v1, v2) are equal (equivalent) if and only if u1 = v1 and u2 = v2 and apply the following operations:

Addition and Subtraction

To add u and v, the corresponding components are added, namely:

u + v               = (u₁ + v₁, u₂ + v₂)

u + v + w       = (u₁ + v₁ + w₁, u₂ + v₂ + w₂)

u – v               = (u₁ – v₁, u₂ – v₂)

Vector multiplication with scalar

To multiply u by scalar k, it is done by multiplying each component with k, namely:

uk = ku = (ku₁, ku₂)

specifically, u = (-u₁, -u₂) dan 0 = 0u = (0,0)

Figure 12 shows that the above definitions are equivalent to the definitions of Geometry that we discussed earlier.

Example 3.

1. If u = (1, -2) and v = 7, 6) then u + v = (1 + 7, -2 + 6) = (8, 4)
2. 4v = 4(7, 6)= (4 (7), 4 (6)) = (28, 24)

Cartesian Coordinate in Space-3

Vectors in space 3 can be expressed by triple real numbers, by introducing the Cartesian coordinate system depicted in Figure 13 below,

If, as Figure 13 above vector v in room 3 is located so that the starting point is at the origin of the Cartesian coordinate system, the coordinates of the terminal point are called components v and we write them as,

v = (v₁, v₂, v₃)

Vector Operations

Two vectors v = (v₁, v₂, v₃) and w = (w₁, w₂, w₃) are equivalent if and only if v₁ = w₁, v₂ = w₂,  and v₃ = w₃.

To add v = (v₁, v₂, v₃) and w = (w₁, w₂, w₃), we add the appropriate components, namely

v + w = (v₁ + w₁, v₂ + w₂, v₃ + w₃)

to multiply vector v with scalar k, we multiply each component with k, i.e.

vk = kv = (k v₁, k v₂, k v₃)

Example 4.

If v = (1, -3, 2) and w = (4, 2, 1) then

v + w = (1 + 4, -3 + 2, 2 + 1)= (5, -1, 3)

2v = (2(1), 2(-3), 2(2))=(2,-6,4)

v-w=v+(-w) = (1 + (-4), -3 + (-2), 2 + (-1)) = (-3, -5, 1)

If the vector has a starting point at P₁ (x₁, y₁, z₁) and the terminal point in P₂(x₂, y₂, z₂), then:

That is, the components  we get by subtracting the coordinates of the starting point from the coordinates of the terminal point. This can be seen using Figure 14 below,

Vector  is the difference between vectors  and  so

Example 5.

Calculate vector components   with the starting point P₁ (2, -1, 4) and terminal point P₂ (7, 5, -5).

Solution:

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