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Thought-Process to Discover Knowledge

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How the property 'direction' affects vector quantities in mathematical calculations?



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The fundamental mathematical operations for vectors are introduced. Let us see how having direction component affects these fundamental operations in vectors.

A person walks `5`m east and then takes the following path

 •  `3`m north

 •  `4.2`m south

 •  `3/4` m north

At this position, how far is the person away from the starting point in the east direction ?

  • `5`m
  • `5`m
  • `5+3+4.2+3/4`m
  • `5+3-4.2+3/4`m
  • none of the above

Answer is '`5`m '– as the person moved `5` meter east and then all his movements were in directions north and south.

Note that any change in north and south direction does not affect the magnitude in east direction.

Any change in a direction affects component along that direction only and does not affect the components in the directions at `90^@` to that direction.

Independence of Quantities along orthogonal directions: For a vector, changes along one axis affect only the component along that axis and do not affect the components along other axes, as the axes are orthogonal.

A ball has a velocity in x-axis `20` m/sec. It also has an unknown velocity in y-axis. In `2` seconds, how far the ball would travel along x-axis?

  • cannot calculate as the velocity along y-axis is not given
  • `20xx2` along x-axis
  • `20xx2` along x-axis

Answer is '`20xx2`' along the x-axis. The velocity causes displacement. The result 'displacement' in one direction is only caused by the component of 'velocity' in that direction.

A person walks `5`m at `30^@` angle to east. Then he turns `90^@` clockwise and walks for 2m. What is the distance from the starting point along the direction `30^@` angle to east?

  • `5`m
  • `5`m
  • `7`m
  • `sqrt(5^2+7^2)`
  • none of the above

Answer is '`5`m' – as the person moved only `5` meter in the direction `30^@` angle to east. He walked in perpendicular to the direction after that.

A person walks `5`m at `30^@` angle to east. Then he turns `40^@` clockwise and walks for 2m. What is the distance from the starting point along the direction `30^@` angle to east?

  • `5`m
  • `7`m
  • `sqrt(5^2+7^2)`
  • none of the above
  • none of the above

Answer is 'none of the above' – The `2`m walk is not at right angle to the direction in question.

This is explained with an illustration in the next page.

Take two persons `A` and `B` start from the point `o` and walks to `p` in direction the `d`.
 •  Person `A` turns at right angle and walks to `q`
 •  Person B turns at an angle `40^@` and walks to `r`

direction components of vector The displacement along direction `d` is asked.
 •  For person `A`, the displacement along `d` is between points `o` and `p`
 •  For person `B` the displacement along `d` is between points `o` and `s`, as the segment `bar(pr)` has component `bar(ps)` along direction `d`.

The component which is in perpendicular to a direction does not affect the component in the direction. The directions which are at `90^@` angle are referred to as orthogonal directions.

In arithmetic operations like addition, subtraction, multiplication by a scalar, and product of vectors, – the results are computed for

  •  components in parallel

  •  components in perpendicular

Each of these two components interact differently.

How does direction affect vector quantities in multiplication? vector multiplication At an abstract level, there are two products possible. Given multiplicand `vec p` and multiplier `vec q`. `vec q` is split into `vec a` and `vec b`, such that
  `vec q = vec a + vec b` and
  `vec a` is in parallel to `vec p`
  `vec b` is in perpendicular to `vec p`

two forms of multiplications are defined for each of these two components.

 •  one with component in parallel to the other, called vector dot product.`vec p cdot vec q = vec p cdot vec a`

 •  another with component in perpendicular to the vector, called vector cross product. `vec p times vec q = vec p times vec b`

Effect of direction: In mathematical calculations, vectors have the following properties
 •  A vector is represented as components along orthogonal directions.

 •  In vector addition, components in parallel add up and components in perpendicular are combined using Pythagoras Theorem.

 •  Two types of vector multiplication are defined,

 •  In vector dot product, only the component of the vector in parallel to the other vector takes part in multiplication.

 •  In vector cross product, only the component of the vector in perpendicular to the other vector takes part in multiplication.

Solved Exercise Problem:

What is repeated addition of a vector?

  • Multiplication of vector by Scalar
  • Multiplication of vector by Scalar
  • Multiplication of magnitude of vector without direction

The answer is 'Multiplication of vector by Scalar'

                            
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