男人资源站-涩涩五月天-加勒比一区二区-欧美丰满少妇-清纯唯美亚洲-蜜臀久久99精品久久久久宅男-1024手机在线观看-a免费在线观看-日韩一区二区三免费高清在线观看-激情欧美一区二区-日韩综合网站-色小说av-夜夜夜操操操-色片在线播放-免费在线观看中文字幕

  • 翰林提供學術活動、國際課程、科研項目一站式留學背景提升服務！
Home » 國際課程 » A-level課程 » Details

Edexcel A Level Further Maths: Core Pure:復習筆記6.1.1 Equations of Lines in 3D

Category: A-level課程, 國際課程, 教材筆記, 福利干貨 Date: 2022年10月20日 上午9:15

Equation of a Line in Vector Form

How do I find the vector equation of a line?

  • You need to know:
    • The?position vector?of one point on the line
    • A?direction vector?of the line (or the position vector of another point)
  • There are two formulas?for getting?a?vector equation?of a line:
    • r?=?a?+?t?(b?-?a)
      • use this formula when you know the position vectors?a?and?b?of two points on the line
    • r?=?a?+?t?d
      • use this formula when you know the position vector?a?of a point on the line and a direction vector?d
    • Both forms could be compared to the Cartesian equation of a 2D line
      • The point on the line?a?is similar to the “+c” part
      • The direction vector?d?or?b?–?a?is similar to the “m” part
  • The vector equation of a line shown above can be applied equally well to vectors in?2 dimensions?and to vectors in?3 dimensions
  • Recall that vectors may be written using ?reference unit vectors?or as?column vectors
  • It follows that in a vector equation of a line either form can be employed – for example,

show the same equation written using the two different forms

How do I determine if a point is on a line?

  • Each?different point?on the line corresponds to a?different value of?t
    • For example: if an equation for a line is?r?= 3i?+ 2j?-?k?+ t (i?+ 2j)
      • the point with coordinates (2, 0, -1) is on the line and corresponds to?t = -1
    • However we know that the point with coordinates (-7, 5, 0) is not on this line
      • No value of?t?could make the?k?component 0

Can two different equations represent the same line?

  • Why do we say?a?direction vector and not?the?direction vector? Because the magnitude of the vector doesn’t matter;?only the direction is important
    • we can multiply any direction vector by a (non-zero) constant and this wouldn’t change the direction
  • Therefore there are an infinite number of options for?a?(a point on the line) and an infinite number of options for the direction vector
  • For Cartesian equations – two equations will represent the same line only if they are multiples of each other
  • For vector equations this is not true – two equations might look different but still represent the same line:

Exam Tip

  • Remember that the vector equation of a line can take many different forms. This means that the answer you derive might look different from the answer in a mark scheme.
  • You can choose whether to write your vector equations of lines using reference unit vectors or as column vectors – use the form that you prefer!
  • If, for example, an exam question uses column vectors, then it is usual to leave the answer in column vectors, but it isn’t essential to do so - you’ll still get the marks!

Worked Example

a) Find a vector equation of a straight line through the points with position vectors?a?= 4i?– 5k?and?b?= 3i?- 3k

al-fm-6-1-1-vector-equation-of-line-we-solution-a

b) Determine whether the point C with coordinate (2, 0, -1) lies on this line.

al-fm-6-1-1-vector-equation-of-line-we-solution-b

 

Equation of a Line in Parametric Form

How do I find the vector equation of a line in parametric form?

  • By considering the three separate components of a vector in the?x, y?and?z?directions it is possible to write the?vector equation?of a line as?three separate equations
    • Letting ?then ?becomes
      • Where ?is a position vector and ?is a direction vector
    • This vector equation can then be split into its three separate component forms:

Worked Example

Write the parametric form of the equation of the line which passes through the point (-2, 1, 0) with direction vector .

al-fm-6-1-1-parametric-equation-of-line-we-solution

Equation of a Line in Cartesian Form

  • The?Cartesian?equation of a line can be found from the?vector equation of a line?by
    • Finding the vector equation of the line in parametric form
    • Eliminating λ?from the parametric equations
      • λ can be eliminated by making it the subject?of each of the parametric equations
      • For example: gives
  • In?2D?the?cartesian?equation of a line?is a regular equation of a straight line simply given in the form
    • by rearranging
  • In?3D?the?cartesian?equation of a line?also includes?z?and is given in the form
    • where
    • This is?given in the formula booklet
  • If one of your variables?does not depend on λ?then this part can be written as a separate equation
    • For example: gives

How do I find the vector equation of a line given the Cartesian form?

  • If you are given the?Cartesian?equation of a line in the form
  • A vector equation of the line can be found by
    • STEP 1: Set each part of the equation equal to λindividually
    • STEP 2: Rearrange each of these three equations (or two if working in 2D) to make?x, y,?and?z?the subjects
      • This will give you the three?parametric equations
    • STEP 3: Write this in the vector form
    • STEP 4: Set?r? to equal
  • If one part of the cartesian equation is given separately and is not in terms of λ?then the corresponding component in the direction vector is equal to zero

Worked Example

A line has the vector equation . Find the Cartesian equation of the line.

al-fm-6-1-1-cartesian-equation-of-line-we-solution-a

 

 

轉載自savemyexams

? 2025. All Rights Reserved. 滬ICP備2023009024號-1

國際競賽 國際競賽
了解背提項目
國際課程 國際課程
國際課程
商務合作 商務合作
商務合作
課程試聽 課程試聽