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

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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`

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