Intersection of lines in R 3 - Problems Theory and Solutions in Linear Algebra

In this section we discuss intersections of lines in R³ and show how to calculate those intersections.

Theoretical Remarks

^2.7.

Given two lines in R³, say ₁ and ₂, we have the following possibilities regarding their intersection:

a) ₁ and ₂ may intersect in a unique common point.

b) ₁ and ₂ may intersect at every point on ₁ and ₂, so that the two lines coincide.

c) ₁ and ₂ may not intersect at any point.

Remark: Given any number of lines inR³, the possibilities of their common intersections are the same as those listed above for two lines.

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Problem

^2.7.1.

Consider the following two lines inR³:

b) Do the lines intersect for b = 1? If so, find the intersection(s) in this case.

Solution

^2.7.1.

a) At any point where ₁and ₂intersect, there must exist parametric values for t and s for the coordinates of the intersection points. To establish those points, we consider

x = 2t + 3 =−s and in matix form we have



The corresponding augmented matrix is



We now apply two elementary row operations to the above augmented matrix, namely

1: multiply the first row by 2 and add the resulting row to the second row;

2: multiply the first row by−1 and add the resulting row to the third row.

This leads to the following echelon form:



From the above echelon form we conclude that the system has a solution for t and s if and only if

b= 2.

Therefore the two lines ₁ and ₂ intersect if and only if b∈ R\{2}.

For those values of b, we have t = 2− 3b

2(b− 2), s = 4 b− 2.

Inserting the above values for t and s into ₁ or ₂, we obtain the x-, y- and z-coordinates of the point of intersection for any b∈ R\{2}, namely

x = −4

Inserting those values for t and s into 1 or 2 we obtain the coordinates of the point of intersection, namely

(4,−1, 3).

Problem

^2.7.2.

Consider the following three lines inR³:

₁:

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a) Find the intersection(s) of the lines ₁ and ₂, if those lines do intersect.

b) Find the intersection(s) of the lines ₁, ₂ and ₃, if those lines do intersect.

Solution

^2.7.2.

a) To find the intersection(s) of ₁ and ₂ we consider x =−t + 3 = 3s + 3

y = 2t + 1 =−6s + 1 z =−t + 2 = 3s + 2, so that

−t − 3s = 0 2t + 6s = 0

−t − 3s = 0.

In matrix form we have



 −1 −3

2 6

−1 −3



 t s



 0 0 0



 .

For the corresponding augmented matrix we have



 −1 −3 0

2 6 0

−1 −3 0



 ∼



 1 3 0 0 0 0 0 0 0



 ,

which means that

t =−3s for all s ∈ R.

Thus for every value of s∈ R for 2 there is a value of t for 1, namely t =−3s, that gives the same coordinates and hence a point of intersection between ₁ and ₂. The two lines, ₁ and ₂, therefore intersect at every point on ₁ (or ₂), so that the two lines in fact coincide.

b) Since ₁ and ₂ coincide (as established above), we can now search for the intersec-tion(s) between 1 and 3. We consider

x =−t + 3 = −4p + 8 y = 2t + 1 = p + 2 z =−t + 2 = 2p − 1,

so that the matrix equation takes the form



 −1 4

2 −1

−1 −2





t p



 5 1

−3



 .

For the corresponding augmented matrix we have



 −1 4 5

2 −1 1

−1 −2 −3



 ∼



 1 −4 −5

0 7 11

0 −6 −8



 ∼



 1 −4 −5

0 1 11/7

0 0 1



 ,

which means that the system is inconsistent. Hence there exist no values for t and p which would give the same point, so that there is no intersection between ₁ and

₃. The three lines, ₁, ₂ and ₃, do therefore not intersect in a common point or points.

2.8 Exercises

1. Consider the matrix equation AX⁻¹+ 3B = A², where

A =

1 −1

0 3

, B =

2 1 1 1

and X is an invertible 2× 2 matrix. Find X, such that the given matrix equation is satisfied.

[Answer: X =

−2/17 −5/17

−1/17 6/17

. ]

2. Consider the matrix equation 2X + AX = 3B,

where A =

1 3

−1 −2

and X is an unspecified matrix.

a) For the given matrix equation, assume that B =

3 1

−3 2

and find the matrix X.

[Answer: X =

9 −6

−6 7

. ]

b) For the given matrix equation, assume that B =

1 −1 −2

−1 1 3

and find the matrix X.

[Answer: X =

3 −3 −9

−2 2 7

. ]

3. Consider the matrix equation C⁻¹(XB− A)B⁻¹ = X,

where B, C and I_n− C are all n × n invertible matrices. Find the matrix X that satisfies the above equation in terms of the other given matrices and in terms of the identity matrix I_n.

[Answer: X = (I_n− C)⁻¹AB⁻¹. ]

4. Consider the matrix equation AX⁻¹+ (X + B)⁻¹ = X⁻¹,

where A, B, X, X + B, A⁻¹− In and A− In are all n× n invertible matrices.

a) Solve the given matrix equation for X.

[Answer: X = B(A⁻¹− In). ]

b) Solve the given matrix equation for X, where A and B take the following explicit forms:

A =

1 −1

1 0

, B =

2 1 1 1

[Answer: X =

−3 2

−2 1

. ]

5. Consider the following matrix:

A =



 a 2 3

1 0 −1

−1 3 a + 6



 ,

where a is an unspecified real parameter.

a) Find all values of a, such that the matrix A is invertible.

[Answer: a∈ R\{1}. ]

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b) Calculate the inverse of A with a = 0.

6. Find all solutions of the system Ax = b for the given matrix A and vector b as given below. Give also the geometrical interpretation of the solutions where possible.

a) A =

[Answer: The unique solution is given by a point in R³, the coordinates of which are x =

[Answer: The solutions are given by a line in R³ passing through the point (3/2, 0, 1/2) and parallel to the vector (1, 1,−1), i.e. the infinitely many solutions are

[Answer: The solutions are given by a plane in R³ with equation x₁− x2+ 2x₃ = 1, i.e. the infinitely many solutions are

[Answer: The system is inconsistent. That is, the system has no solution. ]

7. Find the intersection of the following two planes inR³: Π1: x− y + 3z = 1

Π₂: x + y + 2z = 10.

Use Maple to sketch the planes inR³(see Appendix A for information about Maple).

[Answer: The two planes intersect along the following line:

 :











x =−5t + 28 y = t

z = 2t− 9 for all t∈ R. ]

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8. Find the intersection of the following three planes in R³: Π₁ : x + 3y− 5z = 0

Π2 : x + 4y− 8z = 0 Π₃ :−2x − 7y + 13z = 0.

Use Maple to sketch the planes inR³ (See Appendix A for information about Maple).

[Answer: The three planes intersect along the following line:

 :











x =−4t y = 3t

z = t for all t∈ R. ]

9. Consider the following three planes in R³: Π₁ : x₁− 4x2+ 7x₃ = 1

Π2 : 3x2− 5x3 = 0

Π₃ : −2x1+ 5x₂− 9x3 = k, where k is an unspecified real parameter.

a) Find all values of k such that the given three planes intersect along a common line and give this line of intersection in parametric form.

[Answer: The three planes intersect along a common line if and only if k =−2, where is given by

 :











x₁=−t/5 + 1 x₂= t

x₃= 3t/5 for all t∈ R. ]

b) For which value(s) of k do the three planes intersect in a unique point.

[Answer: There exists no value of k for which the three planes intersect in a unique point. ]

10. Find all solutions of the following system:

11. Consider the following system:

x₁+ x₃+ 2x₄= 1 2x₁+ kx₂+ x₃+ x₄ = 2 3x2+ x3+ 2x4= 3 x₁+ x₂+ x₄ = 4,

where k is an unspecified real parameter.

a) Find all values of k, such that the given system has a unique solution.

[Answer: k ∈ R\{−7}. ]

b) Find all values of k, such that the given system has infinitely many solutions.

[Answer: There exist no values of k for which the system admits infinitely many solutions. ]

c) Find all values of k, such that the given system is inconsistent.

[Answer: k =−7. ]

d) Find all values of k for which the coefficient matrix of the given system is singular.

[Answer: k =−7. ]

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12. Consider the following system:

x₁+ x₂+ x₃ = a 3x₁+ kx₃ = b x1+ kx2+ x3 = c,

where a, b, c and k are unspecified real parameters.

a) Find all values of k, such that the given system has a unique solution for all real values of a, b and c.

[Answer: k∈ R\{1, 3}. ]

b) Find all values of k and the corresponding conditions on a, b and c, such that the given system is consistent.

[Answer: From part a) above, we know that the system has a unique solution (and is consistent) for all k ∈ R\{1, 3} and all real values of a, b and c. For k = 1 the system has infinitely many solutions (and is consistent) if and only if c = a for all c∈ R. For k = 3 the system has infinitely many solutions (and is consistent) if and only if c = 3a− 2b/3 for all a ∈ R and all b ∈ R. ]

13. Consider the following matrix equation:

1 1

−1 1

−

0 1

α −α

X =

1 2

−1 3

where X is an unspecified 2× 2 matrix. Determine all real values of α, such that the given matrix equation has a unique solution for X.

[Answer: α∈ R\{−2}. ] 14. a) Consider the function

f (x) = ax³+ bx²+ cx + d,

where a, b, c and d are unspecified real parameters. Find the values of these parameters such that the graph y = f (x) is passing through the following points in the xy-plane: {(1, 1), (−1, 1), (2, 2), (−2, 12)}. Use Maple to sketch your obtained function f (x) in the xy-plane (see Appendix A for information about Maple).

[Answer: a =−5/6, b = 2, c = 5/6, d = −1. ]

b) Consider the function

f (x) = a cos(2x) + b(π− x) cos(2x) + cx sin(π − x),

where a, b and c are unspecified real parameters. Find the values of these parameters such that the graph y = f (x) is passing through the following points in the xy-plane: {(−π/2, −3π), (π/2, 0), (3π/2, 5π)}. Use Maple to sketch your obtained function f (x) in the xy-plane (see Appendix A for information about Maple).

[Answer: a =−2π, b = 3, c = −1. ]

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15. Consider the system Ax = b with

where h and k are unspecified real parameters.

a) Find the relation between the parameters h and k, such that the given system Ax = b is consistent.

[Answer: k = 2h for all h∈ R. ]

b) Find all solutions for the given system Ax = b.

[Answer: The system has the unique solution x =

h− 3 6− h

for all h∈ R, where k = 2h. ]

16. Consider the following line in R³:

 :

Find all real values of the parameters a, b and c, such that the line is lying on the plane

17. Consider the following two lines inR³:

₁ :

a) Find all values of b, such that the lines ₁ and ₂ intersect.

[Answer: b∈ R\{2}. ]

b) Do the lines intersect for b = 1? If so, find the point of intersection for this case.

[Answer: Yes, the point of intersection has the coordinates (4,−1, 3). ]

18. Consider the following six planes that describe a parallelepiped at their intersections:

Π₁: x + 2y− z = 1 Π₂: 2x + 4y− 2z = 0 Π₃: −3x − y + 2z = 1 Π₄: −9x − 3y + 6z = 1 Π5: y + z =−1

Π₆: −2y − 2z = 3.

Find the vertices, the volume and the midpoint of this parallelepiped, as well as the hight of the parallelepiped with base face described by Π2.

[Answer: The coordinates of the vertices of the parallelepiped are as follows:

P₁: (−17 12, 11

36, −65 36) P₂: (−7

6, 7 18, −25

18) P3: (−3

2, 1 2, −3

2) P₄: (−7

4, 5 12, −23

12) P₅: (−11

12, −7 36, −47

36) P₆: (−2

3, −1 9, −8

9) P7: (−1, 0, −1) P₈: (−5

4, −1 12, −17

12)

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The volume of the parallelepiped is 1/18 cubic units. The coordinates of the the midpoint Q of the parallelepiped is

Q : (−29 24, 11

72, −101 72 ).

The hight of the parallelepiped with base described by Π₂ is 1/√

6 units. ]

19. Consider the following two planes:

Π₁ : x + 2y− 4z = 2 Π2 : x− z = 5.

Find the equation of the planes Π^∗₁, such that Π^∗₁ is the reflection of the plane Π1

about the plane Π₂.

[Answer: Π^∗₁ : 4x− 2y − z = 23. ] 20. Consider the linear system Ax = b with



 1 3 k k 1 4 1 k k



 , b =



 2

−3



 ,

where k is an unspecified real parameter.

a) Find all values of k, such that the given system admits a unique solution.

[Answer: The system admits a unique solution for all k ∈ R\{−2, 2, 3}. ]

b) Find all values of k, such that the given system admits infinitely many solutions, as well as all values of k for which the system is inconsistent.

[Answer: For k =−2 the system has infinitely many solutions, namely x = t



 2 0 1



 +



 −1 1 0



 for all t ∈ R. For k = 2 as well as for k = 3 the system is inconsistent. ]

c) Find all values of k, such that the coefficient matrix A is singular.

[Answer: The matrix A is singular if and only if det A = 0, that is, A is singular for k∈ {−2, 2, 3}. ]

21. The following three lines, ₁, ₂and ₃, describe a triangle inR³at their intersections:

₁ :











x = 4α− 1 y =−2α + 3

z = 8α− 3 for all α ∈ R

₂ :











x =−3β + 7 y =−β + 4

z =−β + 3 for all β ∈ R

₃ :











x =−δ + 6 y =−2δ + 7

z = 3δ− 4 for all δ ∈ R.

Find the area of this triangle.

[Answer: The area of the triangle is 5√

6 square units. ]

22. Consider the homogeneous system Ax = 0, where

A =







1 5 1 k 2 1 k 1 1 4 1 1 4 1 3 1







and k is an unspecified real parameter.

a) Find all values of k, such that the system admits only the trivial solution and all values of k for which A is invertible.

[Answer: For all k ∈ R\{4 3, 7

5} the system admits only the trivial solution x = (0, 0, 0, 0). The matrix A is also invertible for those values of k. ]

b) Find all values of k, such that the system admits infinitely many solutions.

[Answer: For all k = 4

3 or k = 7

5 the system admits infinitely many solutions.]

Chapter 3 Spanning sets and linearly independent sets

The aim of this chapter:

In this chapter we introduce the following definitions and concepts for a finite set of vectors in Rⁿ: linear combinations of vectors, spanning sets and linearly independent sets of vectors. We apply these concepts to describe, for example, a plane or a line in R³ and to gain a better understanding of linear systems.

3.1 Linear combinations of vectors

In this section we introduce the concept of a linear combination for a finite set of vectors inRⁿ.

Theoretical Remarks

^3.1.

Consider the set S of p vectors S ={u1, u2, . . . , up}, where u_j∈ Rⁿ for j = 1, 2, . . . , p.

1. A linear combination of the vectors from the set S is another vector inRⁿ, namely the vector

c₁u₁+ c₂u₂+· · · + cpu_p∈ Rⁿ

for any fixed choice of the p constants c₁, c₂, . . ., c_p, called the scaling factors of the linear combination. That is, v∈ Rⁿ is a linear combination of the vectors from the set S if there exist scaling factors c₁, c₂, . . . , c_p, such that

v = c₁u₁+ c₂u₂+· · · + cpu_p. 133

2. Consider v ∈ Rⁿ and let A be an m× n matrix. Assume now that v is a linear

Then the matrix-vector product Ax is defined as the linear combination of the set of vectors{a1, a₂, . . . , a_n} with scaling factors x1, x₂, . . . , x_n, i.e.

Ax = x₁a₁+ x₂a₂+· · · + xna_n.

Remark: See also Theoretical Remark 2.1 (3) where the matrix-vector product Ax is discussed.

Problem

^3.1.1.

Consider the following set of five vectors inR³: S ={u1, u₂, u₃, u₄, u₅},

Consider also the vector

v =

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a) Show that v is a linear combination of the vectors in the set S and give the linear combination explicitly.

b) Is v a linear combination of the set of vectors{u1, u₂}? Justify your answer.

c) Let A be an unspecified 3× 3 matrix, such that

Au₁=



 1

−1



 , Au₂ =



 2 1 0



 , Au₃=



 3 1 3





Au₄=



 −8 11

−18



 , Au₅ =



 22

−13 32



 .

Find Av explicitly.

Solution

^3.1.1.

a) We have to show that there exist real constants (scaling factors), c₁, c₂, c₃, c₄ and c₅, such that

v = c1u1+ c2u2+ c3u3+ c4u4+ c5u5.

We write this vector equation in the form of a matrix equation, namely [u₁ u₂ u₃ u₄ u₅] c = v,

With the given vectors u_j, we have



and the following corresponding augmented matrix is



Applying several elementary row operations on this augmented matrix, we obtain its unique reduced echelon form, namely



We conclude that the constants c4 and c5 can be chosen arbitrarily, so we let c₄ = t, c₅ = s,

where t and s are arbitrary real parameters. From the above reduced echelon form we then have

In document Problems Theory and Solutions in Linear Algebra (Page 114-140)