I have a question on the following:The line r = i + ( 1 +...

Deromediqm

Deromediqm

Answered

2022-07-14

I have a question on the following:
The line r = i + ( 1 + 3t )j - ( 3 - 4t )k passes through p 1 = ( 1 , 1 , 3 ) and is parallel to v = 3j + 4k.
But why can we say that this is parallel? And how can I apply this for different problems.
If someon could help me, it would be very much appreciated

Answer & Explanation

Anaya Gregory

Anaya Gregory

Expert

2022-07-15Added 14 answers

r = i ^ + j ^ 3 k ^ + t ( 3 j ^ + 4 k ^ ) , t R represents the equation of a straight line that passes through the point with position vector i ^ + j ^ 3 k ^ (take t=0). Take any two distinct points on the line, say for t 1 t 2 R
r 1 = i ^ + j ^ 3 k ^ + t 1 ( 3 j ^ + 4 k ^ ) r 2 = i ^ + j ^ 3 k ^ + t 2 ( 3 j ^ + 4 k ^ )
Then r 1 r 2 is a vector that is parallel to the line, i.e. the line is parallel to ( t 1 t 2 ) ( 3 j ^ + 4 k ^ ) or put simply, the line is parallel to 3 j ^ + 4 k ^
The information about a vector that is parallel to a line is as important as the information about the normal to a plane. It comes in handy in almost all applications involving lines, such as finding the distance of a point from the line, finding the image of a point with respect to a line, finding the angle between a line and a plane, etc..

Do you have a similar question?

Recalculate according to your conditions!

Ask your question.
Get your answer.

Let our experts help you. Answer in as fast as 15 minutes.

Didn't find what you were looking for?