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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

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Vector and Coordinate Geometry

» Vector Algebra in Coordinate Geometry

→ Vector representation can be used to describe points in coordinate space.

→ `p(x,y,z) harr x i + y j + z k`

→ Distance between origin and the point is the magnitude of the vector.**Vector Algebra can be used to solve probems in coordinate geometry.**

» Coordinate Geometry in Vector Algebra

→ Vectors are directed line segments in coordinate space

→ Magnitude is the distance between initial and terminal points

→ Relative properties of vectors, like, angle between vectors, are preserved**Properties and results of coordinate geometry can be used to solve vector algebra problems**

*plain and simple summary*

nub

*plain and simple summary*

nub

dummy

*A point in coordinate plane is equivalently a vector* with origin as initial point, and the given point as terminal point.

Line segment with an initial point to a terminal point specified is a **directed line segment**.

**A vector is a directed line segment** in coordinate plane.

*simple steps to build the foundation*

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*simple steps to build the foundation*

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In this topic, you will learn about how vector algebra can be used in coordinate geometry and how coordinate geometry principles help in vector algebra problems.

Starting on learning "relationship between vectors and coordinate geometry". ;; In this topic, you will learn about how vector algebra can be used in coordinate geometry and how coordinate geometry principles help in vector algebra problems.

Vectors are measurement of quantities with magnitude and directions.

In the mathematical representation of vectors, the components along 3D coordinates are described individually. What is the representation of `vec p`?

- `ai+bj+ck`
- `sqrt(a^2+b^2+c^2)`
- `a+b+c`
- none of the above

Answer is '`ai+bj+ck`'. If you have trouble with answering this, you have to read mathematical representation of vectors.

Coordinate Geometry is the system of geometry where the position of points on the 3D coordinates are described as ordered pairs of numbers corresponding to the three axes. This aims at studying relations and properties of shapes including points, lines, surfaces, and solids. What is the representation of point '`p`'?

- `(a,b,c)`
- `sqrt(a^2+b^2+c^2)`
- `a+b+c`
- none of the above

Answer is '`(a,b,c)`'. If you have trouble with answering this, you have to read basics of coordinate geometry.

Vector Algebra definition of a vector ending at point `p` is

• `ai+bj+ck`

and Coordinate geometry definition of a point `p` is

• `(a,b,c)`

Are these two similar or different?

- Looks similar
- Looks different

They are similar.

In the coming pages, We will see

• How Vector algebra can be used in coordinate geometry problems, and

• How coordinate geometry principles or results are used to solve vector algebra problems.

What is the distance between point `p(2,3)` and point `q(1,-1)`?

- `sqrt(2^2+3^2)+sqrt(1^2+1^2)`
- `sqrt(2^2+3^2)-sqrt(1^2+1^2)`
- `sqrt((2-1)^2+(3-(-1))^2)`
- `(2-1,3-1)`

Answer is '`sqrt((2-1)^2+(3-(-1))^2)`', as Distance between two points is given by `sqrt((x_1-x_2)^2+(y_1-y_2)^2)`

What is the magnitude of the `vec c` in the figure? Given that `vec a=2i+3j` and `vec b=i-j`.

- `|vec b - vec a|`
- `|(1-2)i+(-1-3)j|`
- `sqrt((1-2)^2+(-1-3)^2)`
- all the above

Answer is 'all the above'.

You are not yet introduced to vector addition and subtraction. But it is intuitive to see from the figure that the `vec c` can be traced as negative of `vec a` and then `vec b`.

With the two problems, it is illustrated that the vector algebra can be used to solve coordinate geometry problems.

What does the word 'initial' mean?

- starting; beginning
- a letter

Answer is 'starting; beginning'.

What does the word 'terminal ' mean?

- end; concluding
- a letter

Answer is 'end; concluding'.

A person walks `color(deepskyblue)(2i+j)` (`2` meter towards east and `1` meter towards north) . Then the person walks `color(coral)(4i+5j)`. What is the distance between the starting and end positions?

- `color(deepskyblue)(2i+j)+color(coral)(4i+5j)`
- `color(deepskyblue)(2i+j)-(color(coral)(4i+5j))`
- `color(deepskyblue)(-2i-j)+color(coral)(4i+5j)`
- `color(deepskyblue)(-2i-j)-(color(coral)(4i+5j))`

Answer is '`color(deepskyblue)(2i+j)+color(coral)(4i+5j)`', which is quite intuitive from the given figure.

Given the points `color(deepskyblue)(p(2,1))` and `color(coral)(q(4,5))`. What is the end point on the 3rd side of the triangle formed by two line segments `bar(op)` and a line parallel and equal in length to `bar(oq)` starting at point `p`?

- `(color(deepskyblue)(2)-color(coral)(4), color(deepskyblue)(1)-color(coral)(5))`
- `(color(deepskyblue)(-2)+color(coral)(4), color(deepskyblue)(-1)+color(coral)(5))`
- `(color(deepskyblue)(2)+color(coral)(4), color(deepskyblue)(1)+color(coral)(5))`
- `(color(deepskyblue)(-2)-color(coral)(4), color(deepskyblue)(-1)-color(coral)(5))`

Answer is '`(color(deepskyblue)(2)+color(coral)(4), color(deepskyblue)(1)+color(coral)(5))`', which is quite evident from the given figure.

With the two problems, it is illustrated that the vectors can be considered as line segments in coordinate geometry. The results of 'geometry' can be used to solve vector algebra problems.

This abstraction is applied for vectors that cannot be considered as line segments. Some of the following examples are abstracted as line segments are

• force

• acceleration

• velocity

• displacement

• electric flux density

• etc.

Even though these quantities are not 'line segments' these are abstracted as line segments and coordinate geometry principles are used.

Given the points `A (3,4,1)` and point `B (2,2,4)`, what represents a line from point `A` to point `B`?

- equivalently a vector `vec(AB)`
- it is not a vector.

The answer is 'equivalently a vector `vec(AB)`'. This is named as directed line segment.

What does the word 'directed' mean?

- having a specified direction
- having drawn on a paper

Answer is 'having a specified direction'. For a directed line segment, the initial position and the terminal point are specified. But, a line segment is a line between two points, without identifying one as initial and the other as terminal.

*comprehensive information for quick review*

Jogger

*comprehensive information for quick review*

Jogger

dummy

**Using Vector Algebra in Coordinate Geometry: ** Vector representation can be used to describe points in Coordinate geometry.

• point `p(x_1,y_1,z_1)` is equivalently a vector `vec p = x_1 i + y_1j+z_1k`

• Distance between origin and the point is the magnitude of a vector.

Vector algebra can be used to solve problems in coordinate geometry.

**Directed Line Segment: ** A line from initial point `A (color(deepskyblue)(x_a, y_a, z_a))` to terminal point `B (color(coral)(x_b, y_b, z_b))` is a directed line segment, and equivalently a vector

• `vec(AB) = (color(coral)(x_b)-color(deepskyblue)(x_a))i+(color(coral)(y_b)-color(deepskyblue)(y_a))j+(color(coral)(z_b)-color(deepskyblue)(z_a))k`

• Point `A` is the initial point.

• Point `B` is the terminal point.

• Length of the line segment is the magnitude of `vec(AB)`.

**Abstraction of Vectors in Coordinate Geometry: ** Vectors can be abstracted as directed line segments.

• Magnitude of a vector is the length of the line segment.

• Relative properties of vectors, like angle between vectors, are preserved in this abstraction.

• Properties or results of coordinate geometry can be used to solve vector algebra problems.

• The abstracted vectors can be equivalently moved to any position on the coordinate plane, with magnitude and direction preserved.

*practice questions to master the knowledge*

Exercise

*practice questions to master the knowledge*

Exercise

Find the vector between points `A (1, -3.2, -1)` and `B (2, 1, 2.1)`

- `vec A = i-3.2j-k`
- `vec B = 2i+j+2.1k`
- `vec(AB)= i+4.2j+3.1k`
- `|vec(AB)|=sqrt(1^2+4.2^2+3.1^2)`

The answer is '`vec(AB)= i+4.2j+3.1k`'

What is the angle between the two forces `vec f_1 = 3i` and `vec f_2 = 4j`?

- `0^@`
- `90^@`
- cannot find angle between vectors

The answer is '`90^@`'. The force can be abstracted as directed line segment in coordinate plane. Thus, `f_1 = (3,0)` and `f_2 = (0,4)`. These points are in `x` and `y` axes and so the angle is `90^@`.

what is the distance between points `O (0,0)` and `P (2,3)`?

- magnitude of `vec(OP)`
- `sqrt(2^2+3^2)`
- both the above

The answer is 'both the above'. Points in coordinate plane can be modeled as vectors and vector algebra can be used.

*your progress details*

Progress

*About you*

Progress

Vectors are measurement of quantities with magnitude and directions. In the mathematical representation of vectors, the components along 3D coordinates are described individually. What is the representation of vector p?

i;j;k

a i + b j + c k

square;root;squared

square root a squared + b squared + c squared

a +; a plus; b plus; b +

a+b+c

none;above

none of the above

The answer is "a i + b j + c k". If you have trouble with answering this, you have to read mathematical representation of vectors.

Coordinate Geometry is the system of geometry where the position of points on the 3D coordinates are described as ordered pairs of numbers corresponding to the three axes. This aims at studying relations and properties of shapes including points, lines, surfaces, and solids. What is the representation of point ' p '?

a;b;c

a, b, c

square;root;squared

square root a squared + b squared + c squared

a +; a plus; b plus; b +

a+b+c

none;above

none of the above

The answer is "a, b, c". If you have trouble with answering this, you have to read basics of coordinate geometry.

Vector Algebra definition of a vector ending at point p is a i+b j+c k ;; and Coordinate geometry definition of a point p is (a,b,c);; Are these two similar or different?

similar

Looks similar

different

Looks different

They are similar.

In the coming pages, We will see ;; How Vector algebra can be used in coordinate geometry problems, and ;; How coordinate geometry principles or results are used to solve vector algebra problems.

What is the distance between point p 2,3 ; and point q 1, minus 1?

1

2

3

4

The answer is "square root 2 minus 2 squared plus 3 minus minus 1 squared. ". as Distance between two points is given by square root x 1 minus x 2 squared plus y 1 minus y 2 squared.

What is the magnitude of the vector c in the figure? Given that vector a=2 i+3 j; and vector b=i minus j.

vector

magnitude of vector b minus vector a

minus;plus

magnitude of 1 minus 2 i plus minus 1 minus 3 j

square;root;squared

square root of 1 minus 2 squared + minus 1 minus 3 squared

all;above

all the above

The answer is "all the above". You are not yet introduced to vector addition and subtraction. But it is intuitive to see from the figure that the vector c can be traced as negative of vector a and then vector b.

With the two problems, it is illustrated that the vector algebra can be used to solve coordinate geometry problems.

A point in coordinate plane is equivalently a vector, with origin as initial point, and the given point as terminal point.

What does the word 'initial' mean?

starting; beginning

starting; beginning

letter

a letter

Answer is 'starting; beginning'.

What does the word 'terminal ' mean?

end; concluding

end; concluding

letter

a letter

Answer is 'end; concluding'.

Using Vector Algebra in Coordinate Geometry: Vector representation can be used to describe points in Coordinate geometry. ;; point p x 1, y 1 , z 1 is equivalently a vector p = x 1 i + y 1 j+z 1k ;; Distance between origin and the point is the magnitude of a vector. ;; Vector algebra can be used to solve problems in coordinate geometry.

A person walks 2 i + j, which is 2 meter towards east and 1 meter towards north; Then the person walks 4 i + 5 j. What is the distance between the starting and end positions?

1

2

3

4

The answer is "2i + j + 4 i + 5 j". Which is quite intuitive from the given figure.

Given the points p 2, 1, and q 4, 5. What is the end point on the third side of the triangle formed by two line segments o p and a line parallel and equal in length to o q starting at point p?

1

2

3

4

The answer is "2 + 4, 1 + 5", which is quite evident from the given figure.

With the two problems, it is illustrated that the vectors can be considered as line segments in coordinate geometry. The results of 'geometry' can be used to solve vector algebra problems.

This abstraction is applied for vectors that cannot be considered as line segments. Some of the following examples are abstracted as line segments are ;; force;; acceleration;; velocity;; displacement;; electric flux density;; et cetera. Even though these quantities are not 'line segments' these are abstracted as line segments and coordinate geometry principles are used.

Given the points A 3, 4, 1, and point 2, 2, 4. What represents a line from point A to point B?

equivalently;a;b

equivalently a vector A B

not;it;is

it is not a vector.

The answer is "equivalently a vector A B". This is named as directed line segment.

What does the word 'directed' mean?

specified;direction

having a specified direction

drawn;paper

having drawn on a paper

Answer is 'having a specified direction'. For a directed line segment, the initial position and the terminal point are specified. But, a line segment is a line between two points, without identifying one as initial and the other as terminal.

Line segment with an initial point to a terminal point specified is a directed line segment

Directed line Segment: A line from initial point A; x a, y a, z a ; to terminal point B; x b, y b, z b; is a directed line segment. And equivalently a vector ;; Vector A B = x b minus x a i + y b minus y a j + z b minus z a k ;; Point A is the initial point. ;; Point B is the terminal point. ;; Length of the line segment is the magnitude of vector A B.

Find the vector between points A 1, minus 3 point 2, minus 1 ; and ; B 2, 1, 2 point 1?

1

2

3

4

The answer is "vector A B = i + 4 point 2 j + 3 point 1 k"

A vector is a directed line segment in coordinate plane.

Abstraction of Vectors in Coordinate Geometry: Vectors can be abstracted as directed line segments. ;; Magnitude of a vector is the length of the line segment. ;; Relative properties of vectors, like angle between vectors, are preserved in this abstraction. ;; Properties or results of coordinate geometry can be used to solve vector algebra problems. ;; The abstracted vectors can be equivalently moved to any position on the coordinate plane, with magnitude and direction preserved.

What is the angle between the two forces vector f 1 = 3i and vector f 2 = 4j?

0

0 degree

90;19

90 degree

cannot;find;angle

cannot find angle between vectors

The answer is "90 degree". The force can be abstracted as directed line segment in coordinate plane. Thus f 1 = 3 0 and f 3 = 0 4. These points are in x and y axes and so the angle is 90 degree.

What is the distance between points 0, (0,0) and P (2, 3)?

magnitude;vector

magnitude of vector o p

sqare;root;squared;plus

square root of 2 squared + 3 squared

both;above

both the above

The answer is 'both the above'. Points in coordinate plane can be modeled as vectors and vector algebra can be used.