Vectors

Vector is a quantity having Direction as well as Magnitude

Vectors

Just like the image above :

Side 1 which is a and Side 2 which is b. If you want to find the Side 3 which is c.

You use the equation a² + b² = c² (Note : This is for Right Triangle)

Example :

a = 3 b = 4 so c = 3² + 4² = 5²

If the triangle is not a Right Triangle use this equation instead :

R =  √((F₁² + F₂²) + 2 * F₁ * F₂ * Cos  Θ)

Example : 

F₁ = 3 F₂ = 4 and Θ = 60

R = √((3² + 4²) + 2 * 3 * 4 * Cos (60))

R = √(25 + 2 * 3 * 2) = √(25 + 12) = √37 which is  ≈ 6.08

Just like the image above :

Side 1 = Side 3 which is a and Side 2 = Side 4 which is b. If you want to find the Side 5 which is c.

You use the equation a + b / b + a = c (Note : This is for parallelogram)

Example ;

a = 3 and b = 1 so c = 3 + 1 = 4

R_min ≤ R ≤ R_max While R_min = F₁ - F₂ and R_max = F₁ + F₂

Now using this Triangle we have this equation :

Sin = y ÷ r and Cos = x ÷ r while Tan = y ÷ x

From this we know that F_x = F * Cos Θ  and F_y = F * Sin Θ

Now to find the Magnitude and Direction of the vectors use this equation :

Magnitude = √ΣF_x² + ΣF_y²

Direction = Tan Θ = ΣF_y ÷ ΣF_x

Example :

F₁ = 5 N with Θ = 30

F₂ = 3 N with Θ = 60

F_1y = 5 Cos(30) ≈ 4.33

F_1x = 5 Sin(30) = 2.5

F_2y = 3 Cos(60) = 1.5

F_2x = 3 Sin(60) ≈ 2.60

Magnitude = √4² + 6.9² ≈ 8

Direction = 6.9 ÷ 4 = 1.725

Dot and Cross Product

Dot Product also called Scalar Product. Geometrically the Dot Product of 2 vectors is Magnitude of one time projection of the second onto the first.

A . B = (A_xI + A_yJ + A_zK) . (B_xI + B_yJ + B_zK)

A . B = A_xB_x + A_yB_y + A_zB_z

Same Angle

I . I = J .J = K . K = (1) (1) (Cos 0) = 1

Different Angle

I . J = J . K = K . I = (1) (1) (Cos 90) = 0

Example :

A = 8m

B = 10m

A . B = A B Cos 37 = 8 10 0.8 = 64m

Cross Product also called Vector Product. Geometrically the Cross Product of 2 vectors is area of parallelogram between them.

A x B = C

A is the first, B is the second and C is the resultant. There is a right hand rule to determine the direction of the resultant. Just imagine that A is your finger, B is palm and C is your thumb.

Example :

If A is going to North and B is going to West. Just point your finger to North and face your palm to the West and see where is your thumb pointing at. Your thumb should be pointing Upwards.

I x I = J x J = K x K = (1) (1) (Sin 0) = 0

         I

K               J  

Now this is the IJK Triangle. If it turns in clockwise the result is positive but if it turns in anti-clockwise the result is negative.

Example ;

A = 4i + 3j - 2k

B = 7i + 2j + 5k

A x B = 8k - 21k - 13k - 14j - 20j - 34j + 15i + 4i + 19i = 38i - 68j - 26k