Find if Parametric equations are perpendicular

[salistoun]

Hi all,

How do you find out if this Parametric equation

x = -2t + 3 ; y = -t - 1 ; z = -3t + 2

Is perpendicular to this parametric equation

x = -2 + 6t ; y = 3 - 6t ; z = -3 - 2t

Thanks
Stephen

 

[haruspex]

The constant terms won't affect it, will they? I.e., can't you replace each line by one parallel to it through the origin, and do that simply by omitting the constant terms?
Then it will be just a matter of checking that the dot product of the two 3-d vectors is zero.

 

[chiro]

Hi all,

How do you find out if this Parametric equation

x = -2t + 3 ; y = -t - 1 ; z = -3t + 2

Is perpendicular to this parametric equation

x = -2 + 6t ; y = 3 - 6t ; z = -3 - 2t

Thanks
Stephen

Hey salistoun.

Do you know how to get the direction vector for both parametric representations? How can you test whether two vectors are orthogonal with the dot product?

 

[HallsofIvy]

I was sorely tempted to say that equations, whether parametric or not, are never perpendicular- "perpendicular" is only defined for geometric objects.

But you mean "are the lines given by these parametric equations perpendicular".

The line given by parametric equations $x= A_1t+ B_1$, $y= C_1t+ D_1$, $z= E_1t+ F_1$ and $x= A_2t+ B_2$, $y= C_2t+ D_2$, $z= E_2t+ F_2$ are perpendicular if amd only if their direction vectors $<A_1, C_1, F_1>$ and $<A_2. C_2, F_2>$ are perpendicular which, again, is true if and only if their dot product, $<A_1, C_1, F_1>\cdot<A_2, C_2, F_2>= A_1A_2+ C_1C_2+ F_1F_2= 0$.

 

[blue_raver22]

To determine if two parametric equations are perpendicular, we can use the dot product of their direction vectors. The direction vectors for the first equation are (-2, -1, -3) and for the second equation are (6, -6, -2). When we take the dot product of these two vectors, we get (-2)(6) + (-1)(-6) + (-3)(-2) = 12 + 6 + 6 = 24. Since the dot product is not equal to 0, we can conclude that these two parametric equations are not perpendicular.