You are here:

- Home
- Science
- Mathematics
- Advanced Math
- Lines and planes in space

Advertisement

QUESTION: Consider the line L :(x+ 10)/(-6)=(y-16)/(-2)= (z+ 3)/6

and the plane pi: 3x -3y + 2z = 4.

(a) Explain why L and pi are parallel.

(b) Of all the lines L_2 inside which are parallel to L, find a vector equation for the one which is closest to L.

ANSWER: Questioner: Nick

Country: Quebec, Canada

Category: Advanced Math

Private: No

Subject: linear algebra

Question: vectors

Consider the line L :(x+ 10)/(-6)=(y-16)/(-2)= (z+ 3)/6

and the plane pi: 3x -3y + 2z = 4.

(a) Explain why L and pi are parallel.

(b) Of all the lines L_2 inside which are parallel to L, find a vector equation for the one which is closest to L.

(x+ 10)/(-6)=(y-16)/(-2)= (z+ 3)/6

a) L: x = -6t + 10; y = -2t + 16; z = 6t - 3

DV = <-6,-2,6>

NV to pi = <3,-3,2>

Dot those.

........................................

b) Pick a point on L, such as P(10,16,-3)

Find the point Q IN pi that is the result of 'dropping the perpendicular' to pi from P. That means, along the DV of pi.

Q = P + s NV

Q = (10,16,-3) + s<3,-3,2>

But Q in pi, so Q must satisfy the equation of pi. That will let you solve for s and thus find the coordinates of Q. Use that, plus the DV for L, to get your vector equation.

[an error occurred while processing this directive]---------- FOLLOW-UP ----------

QUESTION: I'm not quite sure by what you mean by this: "But Q in pi, so Q must satisfy the equation of pi. That will let you solve for s and thus find the coordinates of Q."

What i have done so far is:

Q = (0,0,2)

Q = (10,16,-3) + s<3,-3,2>

(0,0,2) = (10,16,-3) + s<3,-3,2>

(-10,-16,5) = s<3,-2,3>

i don't know if this is the right way of doing the problem. Am i doing it right? I undrstand the part saying that when i find the point Q that i have to dot prodcut it afterwards with the DV of L. Thanks for any help.

QUESTION: I'm not quite sure by what you mean by this: "But Q in pi, so Q must satisfy the equation of pi. That will let you solve for s and thus find the coordinates of Q."

Q is in pi, so Q must satisfy the equation of pi, 3x -3y + 2z = 4.

and x = 10 + 3s, y = 16 - 3s, z = -3 + 2s

Substitute and solve for s. Use that value to write Q, the nearest point to the given point on L, then use this Q and the DV of L to write your equation.

What i have done so far is:

Q = (0,0,2) <<<<<<<<<<<<< what's this?

Q = (10,16,-3) + s<3,-3,2>

(0,0,2) = (10,16,-3) + s<3,-3,2>

(-10,-16,5) = s<3,-2,3>

Advanced Math

Answers by Expert:

I can answer questions in basic to advanced algebra (theory of equations, complex numbers), precalculus (functions, graphs, exponential, logarithmic, and trigonometric functions and identities), basic probability, and finite mathematics, including mathematical induction. I can also try (but not guarantee) to answer questions on Abstract Algebra -- groups, rings, etc. and Analysis -- sequences, limits, continuity. I won't understand specialized engineering or business jargon.

I taught at a two-year college for 25 years, including all subjects from algebra to third-semester calculus.**Education/Credentials**

-----------