MYTUTOR SUBJECT ANSWERS

418 views

Two lines have equations r = (1,4,1)+s(-1,2,2) and r = (2,8,2)+t(1,3,5). Show that these lines are skew.

Recall that for two lines to be skew they must satisfy two conditions:

1) They must not be parallel.

2) They must not intersect.

We shall check each condition individually. 

Condition 1

The general vector equation of a line is given by

r = a + kb,

where a is the position vector of some point on the line, k is a scalar, and b is the direction vector of the line. The direction vector of the line, as the name suggests, dictates in what direction the line travels; b tells us how the line is orientated in space. 

For two lines to be parallel, the direction vector of one line must be equal to some scalar multiple of the second line. However, for our two lines, it is clear that there exists no scalar k for which

(-1,2,2) = k(1,3,5).

Thus, the two lines cannot be parallel.

Condition 2

Let us assume that the two lines do in fact intersect. In other words, that

(1,4,1)+s(-1,2,2) = (2,8,2)+t(1,3,5)

for some numbers s and t.

This vector equation leads to three simultaneous equations:

1-s = 2+t   (1), 4+2s = 8+3t   (2), 1+2s = 2+5t   (3).

If we add 2 times Eq. (1) to Eq. (2), we get that

t = -6/5.

If we substitute this value of t into, say, Eq. (3), we get that

s = -5/2.

However, subsituting both of these values into  Eq. (2) yields a contradiction. The LHS gives

4+2(-5/2) = 4-5 = -1,

whereas the RHS gives

8+3(-6/5) = 22/5.

Clearly, then, the LHS is not equal to the RHS; the system of equations is inconsistent, and so the lines do not intersect.

We have shown that the given lines satisfy both of the necessary conditions to be classified as skew. The lines are therefore skew, as required.

Dorian A. A Level Physics tutor, A Level Maths tutor, A Level Further...

1 year ago

Answered by Dorian, an A Level Maths tutor with MyTutor

Still stuck? Get one-to-one help from a personally interviewed subject specialist

176 SUBJECT SPECIALISTS

£24 /hr

Josh R.

Degree: Mathematics (Masters) - Warwick University

Subjects offered: Maths, Further Mathematics

Maths
Further Mathematics

“I study Maths at Warwick, I hope to be able to teach your child to love Maths and how to achieve top exam grades. ”

£20 /hr

James K.

Degree: Law with Accounting and Finance (Bachelors) - Liverpool University

Subjects offered: Maths, Law+ 1 more

Maths
Law
Business Studies

“About Me: I am currently studying Law with Accounting and Finance (LLB Hons) at the University of Liverpool. I have always had a real passion and love for Maths, Economics, Business and reccently discovered Law and hope that my tutori...”

MyTutor guarantee

£20 /hr

Will W.

Degree: Mathematics (Bachelors) - Durham University

Subjects offered: Maths, Physics+ 2 more

Maths
Physics
Further Mathematics
.STEP.

“About Me: I am a Maths and Physics student at Durham University, who has always enjoyed understanding complex concepts in these subjects. I also have a passion for sharing this enjoyment. At my previous school, I ran a Maths Club. Thi...”

About the author

£24 /hr

Dorian A.

Degree: Theoretical Physics (Masters) - Durham University

Subjects offered: Maths, Physics+ 1 more

Maths
Physics
Further Mathematics

“About Me As a Theoretical Physics student at Durham University, I am more than aware of all of the confusing turns that science can take. I have areal passion for my subject, and hope to show my students howbeautiful science can be.  ...”

You may also like...

Posts by Dorian

A satellite is in a stationary orbit above a planet of mass 8.9 x 10^25 kg and period of rotation 1.2 x 10^5 s. Calculate the radius of the satellite's orbit from the centre of the planet.

Two lines have equations r = (1,4,1)+s(-1,2,2) and r = (2,8,2)+t(1,3,5). Show that these lines are skew.

Use De Moivre's Theorem to show that if z = cos(q)+isin(q), then (z^n)+(z^-n) = 2cos(nq) and (z^n)-(z^-n)=2isin(nq).

Other A Level Maths questions

Solve the equation 5^x = 8, giving your answer to 3 significant figures.

If (m+8)(x^2)+m=7-8x has two real roots show that (m+9)(m-8)<0 where m is an arbitrary constant

What is a stable solution and what is dominance?

It is given f(x)=(19x-2)/((5-x)(1+6x)) can be expressed A/(5-x)+B/(1+6x) where A and B are integers. i) Find A and B ii) Show the integral of this from 0 to 4 = Kln5

View A Level Maths tutors

Cookies:

We use cookies to improve our service. By continuing to use this website, we'll assume that you're OK with this. Dismiss

mtw:mercury1:status:ok