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

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Vectors are referred with given names based on the properties:

what does 'null' mean?

- zero; nothing
- 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 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+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.

• **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 `0i+0j+0k`.

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

*Solved Exercise Problem: *

What is `0i` called?

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

The answer is 'all the above'.

what does 'proper' mean?

- correct type or form
- 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
- vectors with direction

The answer is 'vectors with direction'.

• **Proper Vector** (has a direction)

**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`

what does 'unit' measure mean?

- measured `1`
- 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 magnitude `1`
- a vector having angle `1` degree
- 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 `vec p = ai+bj+ck` is an unit vector, then

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

*Solved Exercise Problem: *

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

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

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

what does 'Equal' mean?

- being same in quantity or value
- 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
- 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 `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`

*Solved Exercise Problem: *

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
- all the above

The answer is 'all the above'

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

- having same characteristics or properties
- 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
- 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 `vec p` and `vec q` are like vectors, if

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

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

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

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

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

*Solved Exercise Problem: *

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
- all the above

The answer is 'all the above'

what does 'initial' mean?

- beginning or starting
- beginning or starting
- end or finish

The answer is 'beginning or starting'

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

- jointly; mutually
- 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
- 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 `vec p` and `vec q` are co-initial vectors when they are positioned at the same starting point `(x, y, z)`.

*Solved Exercise Problem: *

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.
- 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?

- of lying in the same line
- 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 on the same line
- 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 `vec p` and `vec q` are collinear vectors if `vec p = n vec q` where `n in RR`.

*Solved Exercise Problem: *

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
- 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.

what does 'co-planar' mean?

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

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

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

**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** (not lying on same plane)

**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`.

*Solved Exercise Problem: *

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

- Yes, they lie in the xy-plane
- 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'.

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

- opposite; reverse
- 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
- both the above

The answer is 'both the above'

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

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

*Solved Exercise Problem: *

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

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

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

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

- constituent part of a larger whole
- 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
- 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'

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

**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.

*Solved Exercise Problem: *

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

- `i`
- `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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