How is the dot product used to find perpendicular lines?

[qazxsw11111]

Hi everyone! So I was looking at vectors and there is this topic which teaches you about using the dot product to find perpendicular lines.

In the sense where AB // CD, AB.CD=0 right?

However, the examples given in the book only uses the directional vector.

E.g. You are given this line l₁. Then you need to find the foot of the perpendicular from a point P to l₁. So the example use the fact that F (foot of perpendicular) lies on l₁ (which l₁ vector parametric equation is given) to get the equation of OF and subsequently PF. However, they dot multiplied the PF with the direction vector of l₁ and not the equation of l₁ which is what I initially thought of.

I know I am misunderstanding something but not sure what it is.

Thank you so much for any clarifications given.

 

[HallsofIvy]

Hi everyone! So I was looking at vectors and there is this topic which teaches you about using the dot product to find perpendicular lines.

In the sense where AB // CD, AB.CD=0 right?

NO! If AB is parallel to CD then AB.CD is equal to the product of the lengths of AB and CD. AB.CD= 0 if AB and CD are perpendicular not parallel.

However, the examples given in the book only uses the directional vector.

E.g. You are given this line l₁. Then you need to find the foot of the perpendicular from a point P to l₁. So the example use the fact that F (foot of perpendicular) lies on l₁ (which l₁ vector parametric equation is given) to get the equation of OF and subsequently PF. However, they dot multiplied the PF with the direction vector of l₁ and not the equation of l₁ which is what I initially thought of.

What do you mean by the dot product of lines anyway?

I know I am misunderstanding something but not sure what it is.

Thank you so much for any clarifications given.

I don't understand what your misunderstanding is! You seem to be complaining that they do not take the dot product of a vector with an equation which would make no sense at all. The dot product is defined only for vectors. Those are the only things you can take the dot product of!