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SCALARS AND VECTORS

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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.

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

A

A

A

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.

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).

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.

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.

- 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.

- 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.

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

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

a

2a

3a

- 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

- 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