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Thought-Process to Discover Knowledge

Welcome to **nub****trek**.

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.continue

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.continue

Welcome to **nub****trek**.

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nubtrek is designed to explain mathematics and science for young readers. Every topic consists of four sections.

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Classification of Vectors

» Classification by magnitude

→ null or zero vector : magnitude `0`

→ unit vector : magnitude `1`

→ proper vector : magnitude not `0`

» By similarities of two vectors

→ equal vectors : All corresponding components equal

→ like vectors: same directions

→ unlike vectors : different directions

→ co-initial vectors : same initial point

→ co-linear vectors : on the same line

→ co-planar vectors : on the same plane

→ non-co-planar vectors : not on the same plane

*plain and simple summary*

nub

*plain and simple summary*

nub

dummy

Vectors are referred with given names based on the properties:

• **Zero ** or ** Null Vector** (magnitude `0`)

• **Proper Vector** (has a direction)

• **Unit Vector** (magnitude `1`)

• **Equal vectors** (same magnitude and direction)

• **Like vectors** (of same direction)

• **Unlike vectors** (of different direction)

• **co-initial vectors** (starting from same position)

• **collinear vectors** (lying on the same line)

• **co-planar vectors** (lying on same plane)

• **non-co-planar vectors** (not lying on same plane)

**Negative** of a Vector (direction reversed)

**Component Form** of a Vector gives the components along three axes.

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What are co-initial, coplanar, collinear vectors? Learn such properties in this topic.

Starting on learning "Different types of vectors". ;; What are co-initial, coplanar, collinear vectors? Learn such properties in this topic.

what does 'null' mean?

- zero; nothing
- not dull

The answer is 'zero; nothing'

Can you make a guess, what is a 'zero or null vector'?

- A vector with zero magnitude
- a vector along one of the axes

The answer is 'A vector with zero magnitude'

Which of the following vector has `0` magnitude?

- `0i+0j+0k`
- `0i+0j`
- `0`
- all the above

The answer is 'all the above'. `0` is a scalar as well as a vector. It is called improper vector as it does not have a direction.

what does 'proper' mean?

- correct type or form
- short form of property

The answer is 'correct type or form'

what is a 'proper vector'?

- vectors without direction
- vectors with direction

The answer is 'vectors with direction'.

what does 'unit' measure mean?

- measured `1`
- measured in isolation

The answer is 'measured `1`'

Can you make a guess, what is an 'unit vector'?

- a vector having magnitude `1`
- a vector having angle `1` degree
- a vector having angle `1` radian

The answer is 'a vector having magnitude `1`'

what does 'Equal' mean?

- being same in quantity or value
- being different in quantity or value

The answer is 'being same in quantity or value'

Can you make a guess when two vectors are 'equal'?

- when the magnitude is same
- when the direction is same
- when both the magnitude and direction are same

The answer is 'when the magnitude and direction are same'

In the following which one is a meaning of 'like'?

- having same characteristics or properties
- attracted to; want

The answer is 'having same characteristics or properties'

Can you make a guess when are two vectors called 'like vectors'?

- when the vectors are perpendicular
- when the vectors have same direction

The answer is 'when the vectors have same direction'.

what does 'initial' mean?

- beginning or starting
- end or finish

The answer is 'beginning or starting'

what does the prefix 'co' mean in co-initial?

- jointly; mutually
- separately; unrelated

The answer is 'jointly; mutually'.

Can you make a guess when two vectors are 'co-initial' vectors?

- when the vectors end at the same position
- When the vectors start from the same position

The answer is 'When the vectors start from the same position'

A vector can be positioned at any point without modifying the defining parameters magnitude and direction. When vectors are used to represent shapes or quantities, the position of the vector is additionally specified.

what does 'collinear' mean?

- of lying in the same line
- of in ascending or descending order

The answer is 'of lying in the same line'. "co" means "together; jointly" ; and "linear" means "line".

Can you make a guess when two vectors are called 'collinear' vectors?

- When the vectors are on the same line
- When the vectors are arranged in magnitude

The answer is 'When the vectors are on the same line'

what does 'co-planar' mean?

- of lying in the same plane
- of having same magnitude and direction

The answer is 'of lying in the same plane'

Can you make a guess when two vectors are 'coplanar'?

- When two vectors are in the same plane
- when two vectors have same magnitude and direction

The answer is 'When two vectors are in the same plane'

which one in the choices is one of the meanings of 'negative'?

- opposite; reverse
- optimistic; confident

The answer is 'opposite; reverse'

What is the negative of `vec p = ai+bj`?

- `-vec p`
- `-ai-bj`
- both the above

The answer is 'both the above'

In the following which one is the meaning of 'component'?

- constituent part of a larger whole
- property of a company

The answer is 'constituent part of a larger whole'

What are the components of a vector?

- x, y, and z components along the three axes
- initial and terminal points of the vector

The answer is 'x, y, and z components along the three axes'

*comprehensive information for quick review*

Jogger

*comprehensive information for quick review*

Jogger

dummy

**Null Vector or Zero Vector: ** A quantity of zero magnitude, given as `vec 0` or `0`. For calculations, it can be used as `0i+0j+0k`.

Technically a null vector is not a vector. Arithmetic operations on vectors like addition, may result in a null vector.

**Proper Vector: ** A vector with non-zero magnitude.

If vector `vec p = ai+bj+ck` is a proper vector then `sqrt(a^2+b^2+c^2) !=0`

**Unit Vector: ** A vector with magnitude 1.

If vector `vec p = ai+bj+ck` is an unit vector, then

`sqrt(a^2+b^2+c^2) = 1`

**Equal Vectors: **The two vectors `vec p = p_x i+p_yj+p_zk` and `vec q = q_x i+q_yj+q_zk` are Equal

`vec p = vec q`

if and only if

`p_x = q_x`

`p_y = q_y`

`p_z = q_z`

**Like Vectors: ** Vectors of same direction.

The vectors `vec p` and `vec q` are like vectors, if

`(vec p)/(|p|) = (vec q)/(|q|)`

**Unlike Vectors: ** Vectors of different direction.

The vectors `vec p` and `vec q` are unlike vectors, if

`(vec p)/(|p|) != (vec q)/(|q|)`

**Co-initial Vectors:**Two vectors `vec p` and `vec q` are co-initial vectors when they are positioned at the same starting point `(x, y, z)`.

**Collinear Vectors: **Two vectors `vec p` and `vec q` are collinear vectors if `vec p = n vec q` where `n in RR`.

**Co-planar Vectors: ** Two vectors `vec p` and `vec q` are coplanar if they lie on the same plane.

Under condition that the positions of vectors are not specified, and the vectors can be equivalently placed anywhere in the 3-D space, any two vectors will be coplanar.

Three vectors `vec p`, `vec q`, `vec r` are co-planar (under the condition that vectors are equivalently positioned anywhere in the 3-D space), if

`|(p_x,p_y,p_z),(q_x,q_y,q_z),(r_x,r_y,r_z)| = 0`

Co-planar property in terms of vector product is given as `vec p cdot (vec q xx vec r) = 0`.

**Non-co-planar Vectors: ** Three vectors `vec p`, `vec q`, `vec r` are non-co-planar (under the condition that vectors are equivalently positioned anywhere in the 3-D space), if

`|(p_x,p_y,p_z),(q_x,q_y,q_z),(r_x,r_y,r_z)| != 0`

Co-planar property in terms of vector product is given as `vec p cdot (vec q xx vec r) != 0`.

**Negative of a Vectors: **For the vector `vec p = ai+bj+ck`, the negative of `vec p` is

`-vec p = -ai-bj-ck`

**Component Form of a Vector: **A vector `vec p` is given in the component form as

`vec p = p_x i + p_yj+p_zk`,

where `p_x, p_y, p_z` are the components along `x, y, z` -axes respectively.

*practice questions to master the knowledge*

Exercise

*practice questions to master the knowledge*

Exercise

What is `0i` called?

- null vector
- improper vector
- zero vector
- all the above

The answer is 'all the above'.

Is `i+j-k` an unit vector?

- Yes, as `1+1-1 = 1`
- No, as magnitude `sqrt(1+1+1) !=1`

The answer is 'No, as magnitude `sqrt(1+1+1) !=1`'

Two vectors `vec p` and `vec q` are equal, then which of the following is true?

- `vec p / (|p|) = vec q / (|q|)`
- directional cosines of `vec p` and `vec q` are equal
- `|p| = |q|`
- all the above

The answer is 'all the above'

Two vectors `vec p` and `vec q` are like vectors, then which of the following is true?

- `vec p / (|p|) = vec q / (|q|)`
- directional cosines of `vec p` and `vec q` are equal
- angles made with `x`, `y`, `z` axes of `vec p` are equal to that of `vec q`.
- all the above

The answer is 'all the above'

Are the vectors `2i+3j` and `4i+6j` co-initial?

- Yes, as one vector is double of the other.
- No, as the vectors are not same.
- Cannot determine from the given information.

The answer is 'Cannot determine from the given information'. The initial position of the vector is to be given separately and when not given, the vectors can be positioned anywhere.

Are all collinear vectors like vectors?

- Yes. Collinear vectors are in same direction
- No. collinear vectors can be either in same direction or in opposite direction

The answer is 'No. collinear vectors can be either in same direction or in opposite direction'.

There are two parameters to note in collinear vectors when comparing them for being like vectors.

1. the direction - whether they are in same direction or in the opposite direction.

2. the position - vectors may be positioned at different points. Like vectors having same direction may not be collinear because of the position.

Are `3i+4j` and `4i-2j` coplanar?

- Yes, they lie in the xy-plane
- No, the information is not sufficient to determine

The answer is 'Yes, they lie in the xy-plane'.

What is the 'negative' of vector `vec p = 2i-j`?

- `-j`
- `-2i-j`
- `-2i+j`
- `2i+j`

The answer is '`-2i+j`'.

What is the component form of unit vector along `x`-axis?

- `i`
- component form cannot be given for unit vectors

The answer is '`i`', in the usual convention of representing component along x-axis using `i`.

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Vectors are referred with given names based on the properties:

what does 'null' mean?

zero;nothing

zero; nothing

not;dull

not dull

The answer is 'zero; nothing'

Can you make a guess, what is a 'zero or null vector'?

zero;0;magnitude

A vector with zero magnitude

along;one;axes

a vector along one of the axes

The answer is 'A vector with zero magnitude'

Which of the following vector has 0 magnitude?

1

2

3

4

The answer is 'all the above'. 0 is a scalar as well as a vector. It is called improper vector as it does not have a direction.

Zero or Null vector : magnitude 0.

Null vector or zero vector: A quantity of zero magnitude, given as vec 0 or 0. For calculations, it can be used as 0 i + 0 j + 0 k. Technically a null vector is not a vector. Arithmetic operations on vectors like addition may result in a null vector.

What is 0i called?

null

null vector

improper

improper vector

zero

zero vector

all;above

all the above

The answer is "all the above"

what does 'proper' mean?

correct;type

correct type or form

short;property

short form of property

The answer is 'correct type or form'

what is a 'proper vector'?

without

vectors without direction

with

vectors with direction

The answer is 'vectors with direction'.

Proper vector : has a direction.

Proper vector: A vector with non-zero magnitude. If vector p = a i + b j + c k is a proper vector then square root of a squared + b squared + c squared does not equal 0.

what does 'unit' measure mean?

1

measured 1

isolation

measured in isolation

The answer is 'measured 1 '

Can you make a guess, what is an 'unit vector'?

magnitude

a vector having magnitude 1

degree

a vector having angle 1 degree

radian

a vector having angle 1 radian

The answer is 'a vector having magnitude 1 '

Unit Vector: magnitude 1.

Unit Vector: A vector with magnitude 1 ;; If vector p = a i + b j + c k is an unit vector, then magnitude = 1.

Is i+j minus k an unit vector?

yes;s;minus

Yes, as 1 + 1 minus 1 = 1

no;magnitude;square;root;not

No, as magnitude square root of 1 plus 1 plus 1 ; not equal 1.

The answer is "No, as magnitude square root of 1 plus 1 plus 1 ; not equal 1."

what does 'Equal' mean?

same

being same in quantity or value

different

being different in quantity or value

The answer is 'being same in quantity or value'

Can you make a guess when two vectors are 'equal'?

magnitude is

when the magnitude is same

direction is

when the direction is same

both

when both the magnitude and direction are same

The answer is 'when the magnitude and direction are same'

Equal vectors: Same magnitude and direction.

Equal Vectors: The two vectors p = p x i + p y j + p z k and vector q = q x i + q y j + q z k are equal if and only if ;; p x = q x ; p y = q y ; and ; p z = q z.

two vectors; vector p and vector q are equal, then which of the following is true?

1

2

3

4

The answer is 'all the above'

In the following which one is a meaning of 'like'?

having;same;characteristics;properties

having same characteristics or properties

attracted;want

attracted to; want

The answer is 'having same characteristics or properties'

Can you make a guess when are two vectors called 'like vectors'?

perpendicular

when the vectors are perpendicular

same;direction

when the vectors have same direction

The answer is 'when the vectors have same direction'.

Like vectors: of same direction.

Like vectors: Vectors of same direction. The vectors p and q are like vectors. if vector p divided by magnitude of p ; equals ; vector q divided by magnitude of q.

Unlike vectors: of different direction.

Unlike vectors: vectors of different direction: The vectors p and q are unlike vectors if vector p divided by magnitude of p ; not equals ; vector q divided by magnitude of q.

Two vector p and q are like vectors, then which of the following is true?

1

2

3

4

The answer is 'all the above'

what does 'initial' mean?

beginning;starting

beginning or starting

end;finish

end or finish

The answer is 'beginning or starting'

what does the prefix 'co' mean in co-initial?

jointly;mutually

jointly; mutually

separately;unrelated

separately; unrelated

The answer is 'jointly; mutually'.

Can you make a guess when two vectors are 'co-initial' vectors?

end;at

when the vectors end at the same position

start;from

When the vectors start from the same position

The answer is 'When the vectors start from the same position'

A vector can be positioned at any point without modifying the defining parameters magnitude and direction. When vectors are used to represent shapes or quantities, the position of the vector is additionally specified.

Co-initial vectors: starting from same position.

Co-initial Vectors: two vectors p and q are co initial vectors when they are position at the same starting point x y z.

Are the vectors 2i+3j and 4i+6j co-initial?

yes;double;other

Yes, as one vector is double of the other.

no;same;not

No, as the vectors are not same.

cannot;determine;from;given;information

Cannot determine from the given information.

The answer is 'Cannot determine from the given information'. The initial position of the vector is to be given separately and when not given, the vectors can be positioned anywhere.

what does 'collinear' mean?

lying;same;line

of lying in the same line

ascending;descending;order

of in ascending or descending order

The answer is 'of lying in the same line'. "co" means "together; jointly" ; and "linear" means "line".

Can you make a guess when two vectors are called 'collinear' vectors?

same;line

When the vectors are on the same line

arranged;magnitude

When the vectors are arranged in magnitude

The answer is 'When the vectors are on the same line'

collinear vectors: lying on the same line.

Collinear Vectors : Two vectors p and q are collinear vectors if vector p equal n times vector q, where n is a real number.

Are all collinear vectors like vectors?

yes;s;same

Yes. Collinear vectors are in same direction

no;either;opposite

No. collinear vectors can be either in same direction or in opposite direction

The answer is "No. collinear vectors can be either in same direction or in opposite direction." ;; There are two parameters to note in collinear vectors when comparing them for being like vectors. ;; first: the direction - whether they are in same direction or in the opposite direction.;; second: the position - vectors may be positioned at different points. Like vectors having same direction may not be collinear because of the position.

what does 'co-planar' mean?

lying;plane

of lying in the same plane

magnitude;direction

of having same magnitude and direction

The answer is 'of lying in the same plane'

Can you make a guess when two vectors are 'coplanar'?

same;plane

When two vectors are in the same plane

magnitude;direction

when two vectors have same magnitude and direction

The answer is 'When two vectors are in the same plane'

co-planar vectors: luing on the same plane.

co planar vectors : two vectors p and q are coplanar, if they lie on the same plane. Under the condition that the positions of vectors are not specified and the vectors can be equivalently placed anywhere in the 3 D space, any two vectors will be coplanar.;; three vectors p , q and r are co planar : under the condition that vectors are equivalently positioned anywhere in the 3 D space. If determinant of p x p y p z ; q x q y q z ; r x r y r z equals 0. ;; Co planar property in terms of vector product is given as p dot q cross r equal 0.

non-co-planar vectors: not lying on same plane.

non-co-planar vectors: three vectors p, q, r are non-co planar if the determinant is not equal 0. Co planar property in terms of vector product is given as p dot q cross r ; not equal; 0

Are 3 i + 4 j and 4 i minus 2 j coplanar?

lie;xy;plane

Yes, they lie in the xy-plane

no;information;not;sufficient;determine

No, the information is not sufficient to determine

The answer is 'Yes, they lie in the xy-plane'.

which one in the choices is one of the meanings of 'negative'?

opposite;reverse

opposite; reverse

optimistic;confident

optimistic; confident

The answer is 'opposite; reverse'

What is the negative of vector p equal a i + b j?

minus vector;- vector; p

minus vector p

minus a; minus b; -a; -b

minus a i minus b j

both;above

both the above

The answer is 'both the above'

Negative of a vector (direction reversed).

-vec p = -ai-bj-ck

What is the negative of vector p = 2 i minus j?

1

2

3

4

The answer is "minus 2 i + j"

In the following which one is the meaning of 'component'?

constituent;par;larger;whole

constituent part of a larger whole

property;company

property of a company

The answer is 'constituent part of a larger whole'

What are the components of a vector?

x;y;z;along;three;3;axes

x, y, and z components along the three axes

initial;terminal;points;vector

initial and terminal points of the vector

The answer is 'x, y, and z components along the three axes'

Component Form of a vector gives the components along three axes.

Component Form of a Vector: A vector p is given in the component form as ; p = p x i + p y j+p z k, ;; where p x, p y, p z are the components along x, y, z -axes respectively.

What is the component form of unit vector along x -axis?

i

i

component;form;cannot;not

component form cannot be given for unit vectors

The answer is ' i ', in the usual convention of representing component along x-axis using i .