TYPES OF VECTOR

Types of Vectors

Zero Vector

We know that all vectors have initial and terminal points. A Zero vector or a null vector is one in which these two points coincide. It is denoted as 0⃗ . Since the magnitude is zero, we cannot assign a direction to these vectors. Alternatively, zero vectors can have any direction. Some examples of zero vectors are AA→, BB→, etc.

Unit Vector

A Unit vector is a vector having a magnitude of unity or 1 unit. A unit vector in the direction of a given vector a⃗  is denoted as a^.

Collinear Vectors

Collinear vectors are two or more vectors which are parallel to the same line irrespective of their magnitudes and direction.

Equal Vectors

If two vectors a⃗  and b⃗  have the same magnitude and direction regardless of the positions of their initial points, then they are Equal vectors. These vectors are written as a⃗  = b⃗ .

Coinitial Vectors

Coinitial vectors are two or more vectors which have the same initial point. For example, AB→ and AC→ are coinitial vectors since they have the same initial point ‘A’.

ALEGEBRA IDENTITIES

Algebra Identities

Difference of Squares

• a2 - b2 = (a-b)(a+b)

• Difference of Cubes

  • a3 - b3 = (a - b)(a2+ ab + b2)

  • Sum of Cubes

    • a3 + b3 = (a + b)(a2 - ab + b2)

    • Special Algebra Expansions

    • Formula for (a+b)2 and (a-b)2

      • (a + b)2 = a2 + 2ab + b2
      • (a - b)2 = a2 - 2ab +b2

      • Formula for (a+b)3 and (a-b)3

        • (a + b)3 = a3 + 3a2b + 3ab2 + b3
        • (a - b)3 = a3 - 3a2b + 3ab2 - b3

        • Roots of Quadratic Equation

        • Formula

          Consider this quadratic equation:

          • ax2 + bx + c = 0

          Where a, b and c are the leading coefficients.

          The roots for this quadratic equation will be:

          • 

CIRCLE

Circle

Definition. Circle is a set of all points in the plane which are equidistant from a given point О, called the center of circle.

Properties of a circle

1. Diameter of circle is equal two radiuses.

D = 2r

2. The shortest distance from the center circle to the secant (chord) is always smaller radius.

3. Three points that not placed on a straight line can hold only one circle.

4. Among all closed curves of equal length, circle has the largest area.

5. If two circle touch at one point, this point placed on the line that passes through the centers of the circles

Area and circumference of circle

Length of circumference

1. Formula of the circumference length in terms of the diameter:

C = πD

2. Formula of the circumference length in terms of the radius:

C = 2πr

Formula of the circle area

1. Formula of the circle area in terms of the radius:

A = πr²

2. Formula of the circle area in terms of the diameter:

A = πD²4

TRIGNOMETRY IDENTITIES

Trigonometry (trig) identities

A. Reciprocal identities

a1	sin  A = 1  csc  A	a4	csc  A = 1  sin  A
a2	cos  A = 1  sec  A	a5	sec  A = 1  cos  A
a3	tan  A = 1  cot  A	a6	cot  A = 1  tan  A

B. Ratio identities

b1	tan  A =  sin  A  cos  A
b2	cot  A =  cos  A  sin  A

C. Opposite Angle identities

c1	sin  ( − A ) = −  sin  A
c2	cos  ( − A ) = c o s A  (Caution - not like c1, c3)
c3	tan  ( − A ) = −  tan  A

D. Complementary angle identities

e1	sin  A =  cos  π 2 − A
e2	tan  A =  cot  π 2 − A
e3	sec  A =  csc  π 2 − A