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mathsVector AlgebraVectors and Coordinate Geometry

### Vector 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.

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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
• 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)
• (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 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)
• 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
• 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.

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
• a letter

What does the word 'terminal ' mean?

• end; concluding
• end; concluding
• a letter

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.

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)
• 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))
• (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)
• 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 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.

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 (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).

Solved Exercise Problem:

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)= 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'

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.

Solved Exercise Problem:

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

• 0^@
• 90^@
• 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^@.

Solved Exercise Problem:

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
• both the above

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

slide-show version coming soon