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3. (a) Show that l₁ given by x = 3 +8 Y = 3+ s Z = -1+28 is not parallel to the line l₂: x(t) = [1, −4, −1] + t[3, -2, 1] but they do not intersect. (b) Find a non-zero vector n that is orthogonal to both lines ₁ and l2. Show that l₁ and l₂ lie in 2 parallel planes. Find general equations of the two parallel planes containing these lines. [Hint: If a line in R³ with direction vector d lies in a plane with normal vector n, then d and n must be orthogonal.]

3. (a) Show that l₁ given by x = 3 +8 Y = 3+ s Z = -1+28 is not parallel to the line l₂: x(t) = [1, −4, −1] + t[3, -2, 1] but they do not intersect. (b) Find a non-zero vector n that is orthogonal to both lines ₁ and l2. Show that l₁ and l₂ lie in 2 parallel planes. Find general equations of the two parallel planes containing these lines. [Hint: If a line in R³ with direction vector d lies in a plane with normal vector n, then d and n must be orthogonal.]

Linear Algebra: A Modern Introduction
Linear Algebra: A Modern Introduction
4th Edition
ISBN:9781285463247
Author:David Poole
Publisher:David Poole
Chapter1: Vectors
Section1.3: Lines And Planes
Problem 18EQ
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3. (a) Show that l₁ given by x = 3 +8 Y = 3+ s Z = -1+28 is not parallel to the line l₂: x(t) = [1, −4, −1] + t[3, -2, 1] but they do not intersect. (b) Find a non-zero vector n that is orthogonal to both lines ₁ and l2. Show that l₁ and l₂ lie in 2 parallel planes. Find general equations of the two parallel planes containing these lines. [Hint: If a line in R³ with direction vector d lies in a plane with normal vector n, then d and n must be orthogonal.]
Transcribed Image Text:3. (a) Show that l₁ given by x = 3 +8 Y = 3+ s Z = -1+28 is not parallel to the line l₂: x(t) = [1, −4, −1] + t[3, -2, 1] but they do not intersect. (b) Find a non-zero vector n that is orthogonal to both lines ₁ and l2. Show that l₁ and l₂ lie in 2 parallel planes. Find general equations of the two parallel planes containing these lines. [Hint: If a line in R³ with direction vector d lies in a plane with normal vector n, then d and n must be orthogonal.]
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