MYTUTOR SUBJECT ANSWERS

164 views

Do the following vector equations intersect? l = (1 + μ)i + (2 - μ)j + (2μ - 5)k, and m = 2λi + 3j + (2 + λ)k.

Firstly we must appreciate that if there is an intersection between l and m, this means that there exists a point that occurs on both lines. Therefore if there is an intersection, it is when l = m.

So we must start by making the two vector equations equal to eachother:

(1 + μ)i + (2 - μ)j +(2μ - 5)k = 2λi + 3j + (2 + λ)k

From here we can equate the coefficients to make 3 simultaneous equations with the variables λ and μ. This means making the coefficients of i equal to eachother, the coefficients of j equal to eachother and the coefficients of k equal to eachother. This gives us the following equations:

1 + μ = 2λ (equation 1),

2 - μ = 3 (equation 2),

2μ - 5 = 2 + λ (equation 3).

From equation 2 we can see that μ = -1. Using this value of μ, we can solve for λ in equation 1 using substitution, which gives:

1 - 1 = 2λ, so 2λ = 0, giving λ = 0.

So now we have a value for λ and a value for μ. Here we can show whether the two vectors intersect or not by substituting these values into equation 3. The two vectors intersect if and only if doing this gives a true statement (e.g 1 + 1 = 2).

So by substituting μ = -1 and λ = 0 into the equation 2μ - 5 = 2 + λ, we get:

2(-1) - 5 = 2 + 0, which is the same as

-7 = 2.

Obviously, this is not a true statement and therefore we can conclude that the two lines do not intersect at any point.

Harry D. A Level Maths tutor

4 months ago

Answered by Harry, an A Level Maths tutor with MyTutor


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

244 SUBJECT SPECIALISTS

£20 /hr

Solomon L.

Degree: Mathematics (Bachelors) - Leeds University

Subjects offered: Maths, Further Mathematics

Maths
Further Mathematics

“Currently studying at Leeds university, ready to help you improve your grades in Maths and Chemistry GCSE! ”

£20 /hr

Killian D.

Degree: Mathematics (Bachelors) - Edinburgh University

Subjects offered: Maths, .MAT.

Maths
.MAT.

“Who am I? I am a first year Mathematics student in the University of Edinburgh. I am studying full time but have chosen to tutor as well as I have a passion for the subject; something I believe you will discover for yourself in our tu...”

MyTutor guarantee

£30 /hr

Brad D.

Degree: Natural Sciences (Masters) - Durham University

Subjects offered: Maths, Physics

Maths
Physics

“About me: Hey everyone, I'm Brad! I'm studying for a Masters degree in Natural Sciences at Durham, specializing inMaths and Physics. Even while studying at school, I loved the sciences, and I loved being able to help people fully come...”

About the author

£20 /hr

Harry D.

Degree: Mathematics with a year abroad (Masters) - Sheffield University

Subjects offered: Maths

Maths

“My name is Harry, I am a Mathematics student at the University of Sheffield. I am very enthusistic about the subject and love to help others improve their skills. I am a well organised, patient and friendly individual and I feel these ...”

MyTutor guarantee

You may also like...

Other A Level Maths questions

C1 - Simplifying a fraction that has a root on the denominator

What is 'Chain Rule' and why is it useful?

How would you differentiate f(x)=3x(2x-1)^2

Why is it that sin^2(x) + cos^2(x) = 1?

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