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Vectors

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A vector has magnitude and direction and is represented by an arrow defining the direction, and the length of the arrow defines the vector's magnitude. The magnitude of a vector is denoted by its length. Vector is a triplet of numbers represented as r = (a, b, c) where a, b and c are the components of r. Two vectors can be added and it is possible to join them head to tail. The vector, as a mathematical object, is defined as a directed line segment. Magnitude of the vector is denoted by B or by simple |B|.
Vectors Math

A vector with a magnitude zero is called as null vector. Two vectors, A and B are equal if they have the same magnitude and direction, regardless of whether they have the same initial points. A vector having the same magnitude as A but in the opposite direction to A is denoted by -A. 

Examples of vector: Velocity, acceleration, displacement, weight, force etc., 

What are Vectors?

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Vector is a quantity having length as well as direction. Vector represents magnitude and direction of a physical quantity. A vector can be represented using an ordered set of components in such a way that

V = ($V_{1},V_{2},........ V_{n}$)
Here $V_{1}, V_{2},......., V_{n}$ are the components of the vector v. 

Vector can be represented in a matrix from as
$\begin{bmatrix}
V_{1}\\
V_{2}\\
.\\
.\\

V_{n}\end{bmatrix}$
where $V_{1}, V_{2}, ......V_{n}$ are the components of the vector v.

Vector Equation

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Using a parameter z we can define a vector coordinate that specify both the x and y coordinates of a graph.

Vector equation of a line segment and plane are written as r = $r_{0}$ + t r$_{1}$ and r(t) = ( 1 - t) r$_{0}$ + t r$_{1}$ respectively, where r$_{0}$, r$_{1}$ are the end points of the vector.
Parametric Equation
x = t and y = 2t

Vector Form
(x, y) = (t, 2t)

Addition of Vectors

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Sum of two vectors A and B is a vector C. Vectors are added according to the triangle law of addition. Let $\vec{a}$ and $\vec{b}$ be two vectors considered for vector addition. Choose an arbitrary point O. Take O as the initial point of $\vec{a}$ and A be its terminal point so that $\vec{OA}$ = $\vec{a}$. Place the initial point of $\vec{b}$ on the terminal point A of $\vec{a}$. Let the terminal point of $\vec{b}$ be B, so that $\vec{AB}$ = $\vec{b}$. Draw vector $\vec{OB}$, which is sum of $\vec{a}$ and $\vec{b}$ and we write $\vec{OB}$ = $\vec{a}$ + $\vec{b}$.

Addition of Vectors
Complete the parallelogram OABC. Clearly OA = CB and $\vec{OA}$ and $\vec{CB}$ they are equal vectors. Similarly $\vec{OC}$ and $\vec{AB}$ are equal vectors.

Vector Graphics

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Vector graphics is a computer image that is stored and displayed in terms of vectors rather than points allowing for easier scaling and storage. It uses geometrical primitives such as points, lines, curves and polygons based on mathematical equations to represent images in computer graphics. Once a vector image is drawn we save what we did. But when you open that image the computer retraces you steps and virtually redraws you picture for you, that is exactly the picture you made it.

Vector Problems

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Solved Examples

Question 1: Find the angle between a = 3i + 2j + 4k and b = 5i + 9j + 2k
Solution:
 
Here we need to find the angle between a and b
So we first find the magnitude of a and b then find their angles.

|a| = $\sqrt{9 + 4 + 16}$ = $\sqrt{29}$
|b| = $\sqrt{25 + 81+ 4}$ = $\sqrt{110}$
a.b = 3.5 + 2.9 + 4.2
= 15 + 18 + 8
= 41
$\theta$ = $cos^{-1}$$({\frac{41}{\sqrt{29}\sqrt{110}}})$
 

Question 2: Find the cross product between a = (2, -3, 5) and b = (4, 2, 3)
Solution:
 
Given a = (2, -3, 5) and b  = (4, 2, 3)

Cross product is a * b = $\begin{bmatrix}
i & j & k\\
 2&-3  & 5\\
 4& 2& 3
\end{bmatrix}$

= i(-9 - 10) - j (6 - 20) + k (4 + 12)

= -19i + 14j + 16k