Vector, parametric, and symmetric equations of the very same line

Vector, parametric, and also symmetric equations are different varieties of equations that can be supplied to stand for the very same line.

You are watching: Find parametric equations and symmetric equations for the line


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We use various equations at various times to tell us information about the line, so we have to know how to find all three varieties of equations.

Thevectorequation of a heat is provided by

???r=r_0+tv???

where ???r_0??? is a point on the line and also ???v??? is a parallel vector

Theparametricequations that a heat are given by

???x=a???

???y=b???

???z=c???

where ???a???,???b??? and???c??? room the coefficients indigenous the vector equation???r=a\bold i+b\bold j+c\bold k???

Thesymmetricequations of a line are provided by

???\fracx-a_1v_1=\fracy-a_2v_2=\fracz-a_3v_3???

where ???a(a_1,a_2,a_3)??? space the works with from a point on the line and also ???v_1???, ???v_2??? and ???v_3??? are the collaborates from a parallel vector.


How come convert in between vector, parametric, and symmetric equations of a line


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Finding the three types of equations that a line the passes through a details point and is perpendicular to a vector equation

Example

Find the vector, parametric and symmetric equations of the line that passes through the point ???a(2,-1,3)??? and is perpendicular come ???2\bold i-\bold j+4\bold k=1???.

Before we get started, we have the right to say the the given allude ???a(2,-1,3)??? can additionally be stood for by ???2\bold i-\bold j+3\bold k???. Additionally, we understand that the offered vector ???2\bold i-\bold j+4\bold k=1??? deserve to be represented by ???\langle2,-1,4\rangle???, or ???2\bold i-\bold j+4\bold k???. To summarize what us know, we have


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We use different equations at different times come tell us information about the line, so we should know just how to find all three types of equations.


To find the vector equation of the line, we’ll use ???r=r_0+tv???, whereby ???r_0??? is the suggest on the heat ???2\bold i-\bold j+3\bold k??? and also ???v??? is the perpendicular vector ???2\bold i-\bold j+4\bold k???.

See more: What Are The Possible Rational Zeros Of F(X) = X4 + 2X3 − 3X2 − 4X + 12? ?

???r=(2\bold i-\bold j+3\bold k)+t(2\bold i-\bold j+4\bold k)???

???r=2\bold i-\bold j+3\bold k+2\bold it-\bold jt+4\bold kt???

???r=(2\bold i+2\bold it)+(-\bold j-\bold jt)+(3\bold k+4\bold kt)???

???r=(2+2t)\bold i+(-1-t)\bold j+(3+4t)\bold k???

With the vector equation of this line in hand, it’ll be really easy for united state to uncover the parametric equations the the line, due to the fact that all we have to do is take the coefficients indigenous the vector equation, and also the parametric equations are

???x=2+2t???

???y=-1-t???

???z=3+4t???

To find the symmetric equations, we’ll just plug the given coordinate allude in for ???a_1???, ???a_2??? and also ???a_3???, add to the coefficients native the perpendicular vector in for ???v_1???, ???v_2???and ???v_3???.