equations-of-lines-in-3d

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,

  and <img alt=”bold r equals open parentheses table row 3 row 1 row cell negative 7 end cell end table close parentheses plus t open parentheses table row 1 row cell negative 2 end cell row 0 end table close parentheses” data-mathml='<math ><semantics><mrow><mi mathvariant=”bold”>r</mi><mo>=</mo><mfenced separators=”|”><mtable><mtr><mtd><mn>3</mn></mtd></mtr><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>7</mn></mtd></mtr></mtable></mfenced><mo>+</mo><mi>t</mi><mfenced separators=”|”><mtable><mtr><mtd><mn>1</mn></mtd></mtr><mtr><mtd><mo>-</mo><mn>2</mn></mtd></mtr><mtr><mtd><mn>0</mn></mtd></mtr></mtable></mfenced></mrow><annotation encoding=”application/vnd.wiris.mtweb-params+json”>{“language”:”en”,”fontFamily”:”Times New Roman”,”fontSize”:”18″}</annotation></semantics></math>’ height=”80″ role=”math” 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