Answer to Question #95116 in Analytic Geometry for Dhiman

Question

Full notes on vector concept

Answer 1

Vector Definition

A vector is an object that has both a magnitude and a direction. Geometrically, we can picturise a vector as a line segment, whose length is the magnitude of the vector and with an arrow indicating the direction. The direction of the vector is from its tail to its head.

$\to$

Two vectors are said to be the same if they have the same magnitude and direction. This means that if we take a vector and move it to a new position (without rotating it), then the vector we obtain at the end of this translation is the same vector we had in the beginning.

Two examples of vectors are those that represent momentum and electric field. 

We denote vectors using arrows as in $\overset{\to}{a}$

We denote the magnitude of the vector $\overset{\to}{a}\ by\ ||a||$

Vectors in the plane

To determine the coordinates of a vector a

a in the plane, the first step is to translate the vector so that its tail is at the origin of the coordinate system. Then, the head of the vector will be at some point $(a_1,a_2)$ in the plane.We call 

$(a_1,a_2)$  the coordinates or the components of the vector $\overset{\to}{a}$ We often write $a\in\ R^2$

to denote that it can be described by two real coordinates.

Using the Pythagorean Theorem, we can obtain an expression for the magnitude of a vector in terms of its components. Given a vector $\overset{\to}{a}=(a_1,a_2)$

the vector is the hypotenuse of a right triangle whose legs are $a_1 \ and \ a_2$ ,

Hencethe length of the vector is

$||a||=\sqrt{{a_1}^2+{a_2}^2}$

Vectors in three-dimensional space

In three-dimensional space, there is a standard Cartesian coordinate system (x,y,z). Starting with a point which we call the origin, construct three mutually perpendicular axes, which we call the 

x-axis, y-axis and the z-axis. 

With these axes any point p can be assigned three coordinates $p=(p_1,p_2,p_3)$

Just as in two-dimensions, we assign coordinates of a vector $\overset{\to}{a}$ by translating its tail to the origin and finding the coordinates of the point at its head. In this way, we can write the vector as .

$a=(a_1,a_2,a_3)$ We often write $a\isin R^3$

to denote that it can be described by three real coordinates. Sums, differences, and scalar multiples of three-dimensional vectors are all performed on each component. 

Therefore,by Pythagoras theorem we can write

$||a||=\sqrt{a_1^2+a_2^2+a_3^2}$

Reference-https://mathinsight.org/vector_introduction