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Thought-Process to Discover Knowledge

Welcome to nubtrek.

Books and other education websites provide "matter-of-fact" knowledge. Instead, nubtrek provides a thought-process to discover knowledge.

In each of the topic, the outline of the thought-process, for that topic, is provided for learners and educators.

Read in the blogs more about the unique learning experience at nubtrek.
mathsVector AlgebraMultiplication of Vectors by Scalar

### Component form of Vectors

Vectors are represented in component form as sum of standard unit vectors multiplied by scalars.

click on the content to continue..

If a person walks 3 meter north east, 4 meter north. Then, facing north, the person turns clockwise 120^@ and walk 2 meter forward. What is the distance he has to walk to return back to the starting position?

• Answer is 3+4+2
• answer is not 3+4+2
• answer is not 3+4+2

The answer is 'not 3+4+2'. The figure shows the problem using line segments and to solve this, trigonometry is used.

Represent each of the segment in their horizontal and vertical components.
•  3 meter north east = 3cos45 meter east + 3sin45 meter north

Represent each of the segment in their horizontal and vertical components.
•  3 north east = 3cos45 east + 3sin45 north
•  4 north = 4cos90 east + 4sin90 north
•  2 at -30^@ = 2cos(−30) east + 2sin(−30) north
•  Add each of these east and north components individually
•  3cos45+4cos90+2cos(−30) = 3.85
•  3sin45+4sin90+2sin(−30) = 5.12
•  result distance  = 3.85i+5.12j

[Continued..] What made the solution simple?

• The vector quantities are split into components having same direction
• The vector quantities are split into components having same direction
• The vector quantities are split into components having same magnitude

The answer is 'the vector quantities are split into components having same direction' which were easily added.

Note that a vector is given as a sum of components along three dimensions, which is equivalently a sum of scalar multiples of the three unit vectors.

Vector representation is chosen to be sum of standard unit vectors multiplied by scalars.

Vector Component form: Component form of vector representation is the sum of standard unit vector i, j, k multiplied by scalars a, b, c in RR.
vec p = a i + bj + ck

slide-show version coming soon