Scalars and vectors
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SCALARS AND VECTORS. SCALARS AND VECTORS. Scalar is a simple physical quantity that is not changed by coordinate system rotations or translations. Expressing a scalar quantity we give it simply with a number and a unit (for example, 12 kg) .

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SCALARS AND VECTORS

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Scalars and vectors

SCALARS AND VECTORS


Scalars and vectors1

SCALARS AND VECTORS

  • Scalar is a simple physical quantity that is not changed by coordinate system rotations or translations.

  • Expressing a scalar quantity we give it simply with a number and a unit (for example, 12 kg).

  • If a quantity has both a magnitude and direction, it is called a vector.


Scalars and vectors2

SCALARS AND VECTORS


Vectors

Vectors

  • Vectors are equal when they have the same magnitude and direction, irrespective of their point of origin.

A

A

A


Sum of two vectors

SUM OF TWO VECTORS

C= A + B

B

A

B

If two vectors have the same direction, their resultant has a magnitude equal to the sum of their magnitudes and will also have the same direction.


Sum of two vectors1

SUM OF TWO VECTORS

C= A + B

A

B

B

If two vectors have the opposite direction, their resultant has a magnitude equal to the subtraction of their magnitudes; direction of the sum is equivalent to the direction of longer vector (to the bigger magnitude).


Sum of two vectors by a graphical method

sum of two vectors by a graphical method

B

A

C=A+B

B

Two vectors A and B are added by drawing the arrows which represent the vectors in such a way that the initial point of B is on the terminal point of A. The resultant C = A + B, is the vector from the initial point of A to the terminal point of B.


Polygon method

Polygon method


Parallelogram method

Parallelogram method

In the parallelogram method for vector addition, the vectors are translated, (i.e., moved) to a common origin. The resultant R is the diagonal of the parallelogram drawn from the common origin.


Method of components

Method of components

  • The components of a vector are those vectors which, when added together, give the original vector.

  • The sum of the components of two vectors is equal to the sum of these two vectors.


Rectangular components

Rectangular components

  • In all vector problems a natural system of axes presents itself. In many cases the axes are at right angles to one another.

  • Components parallel to the axes of a rectangular system of axes are called rectangular components.

  • In general it is convenient to call the horizontal axis X and the vertical axis Y.

  • The direction of a vector is given as an angle counter-clockwise from the X-axis.


Rectangular components1

Rectangular components

  • The magnitude of A,|A|, can be calculated from the components, using the Theorem of Pythagoras:

  • and the direction can be calculated using


Multiplication of vectors by positive scalar

Multiplication of vectors bypositive scalar

  • Scalar multiplication of vector by a positive real number multiplies the magnitude of the vector without changing its direction.

a

2a

3a


Multiplication of vectors by negative scalar

Multiplication of vectors by negative scalar

  • Scalar multiplication of vector by a negative real number multiplies the magnitude of the vector and changes its direction into opposite directions.

a

-2a

-3a


D ivision of vectors by scalars

Division of vectors by scalars

  • Scalar dividing of vector by a real number is equal to multiplication with reciprocal (or multiplicative inverse) of that number.

a÷2 = ½ x a

a

b÷(-3) = -1/3 x b

b


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