Find the distance from (3, 2, 5) to the line in 11

The first equation in (2.6) written out in detail is M11x1 + M12x2 + M13x3 = k1. Write out the other two equations in the same way and then substitute x1, x2, x3 = x, y, z and the values of Mij and ki from (2.4) and (2.5) to verify that (2.6) is really (2.3).

As in Problem 1, write out in detail in terms of Mij , xj , and ki, equations like (2.6) for two equations in four unknowns; for four equations in two unknowns.

x 2y + 13 = 0 y 4x = 17

2x + y z = 2 4x + y 2z = 3

2x + y z = 2 4x + 2y 2z = 3

x + y z = 1 3x + 2y 2z = 3

8 < : 2x + 3y = 1 x + 2y = 2 x + 3y = 5

x + y z = 4 x y + 2z = 3 2x 2y + 4z = 6

8 < : x y + 2z = 5 2x + 3y z = 4 2x 2y + 4z = 6

8 < : x + 2y z = 1 2x + 3y 2z = 1 3x + 4y 3z = 4

8 < : x 2y = 4 5x + z = 7 x + 2y z = 3

8 < : 2x + 5y + z = 2 x + y + 2z = 1 x + 5z = 3

< : 4x + 6y 12z = 7 5x 2y + 4z = 15 3x + 4y 8z = 4

8 < : 2x + 3y z = 2 x + 2y z = 4 4x + 7y 3z = 11

@ 112 246 325

0 @ 2 353 4 111 3 234 1 A

BB@ 11 4 3 3 1 10 7 4 2 14 10 20 6 4

0 BB@ 1 0 10 1 2 1 0 2 2 53 2 4 86 1 CCA

Find a vector perpendicular to both i 3j + 2k and 5i j 4k.

Find a vector perpendicular to both i + j and i 2k.

Show that B|A| + A|B| and A|B| B|A| are orthogonal.

Square (A + B); interpret your result geometrically. Hint: Your answer is a law which you learned in trigonometry

If A = 2i 3j + k and A B = 0, does it follow that B = 0? (Either prove that it does or give a specific example to show that it doesnt.) Answer the same question if A B = 0. And again answer the same question if A B = 0 and A B = 0.

What is the value of (A B) 2 + (A B) 2? Comment: This is a special case of Lagranges identity. (See Chapter 6, Problem 3.12b, page 284.)

The sum of the squares of the diagonals of a parallelogram is equal to twice the sum of the squares of two adjacent sides of the parallelogram.

The median to the base of an isosceles triangle is perpendicular to the base.

In a kite (four-sided figure made up of two pairs of equal adjacent sides), the diagonals are perpendicular.

The diagonals of a rhombus (four-sided figure with all sides of equal length) are perpendicular and bisect each other.

Find a point on the plane 2x y z = 13. Find the distance from (7, 1, 2) to the plane.

Find the distance from the origin to the plane 3x 2y 6z = 7.

Find the distance from (2, 4, 5) to the plane 2x + 6y 3z = 10.

Find the distance from (3, 1, 2) to the plane 5x y z = 4.

Find the perpendicular distance between the two parallel lines in Problem 12.

Find the distance (perpendicular is understood) between the two parallel lines in Problem 13.

Find the distance from (2, 5, 1) to the line in Problem 10.

Find the distance from (3, 2, 5) to the line in Problem 11

Determine whether the lines x 1 2 = y + 3 1 = z 4 3 and x + 3 4 = y + 4 1 = 8 z 4 intersect. Two suggestions: (1) Can you find the intersection point, if any? (2) Consider the distance between the lines.

Find the angle between the lines in Problem 37.

r = 2j + k + (3i k)t1 and r = 7i + 2k + (2i j + k)t2.

r = (5, 2, 0) + (1, 1, 1)t1 and r = (4, 4, 1) + (0, 3, 2)t2.

r = (4, 3, 1) + (1, 1, 1)t and r = (4, 1, 1) + (1, 2, 1)t.

The line that joins (0, 0, 0) to (1, 2, 1), and the line that joins (1, 1, 1) to (2, 3, 4).

\(\frac{x-1}{2}=\frac{y+2}{3}=\frac{2 z-1}{4}\) and \(\frac{x+2}{-1}=\frac{2-y}{2}\), \(z=\frac{1}{2}\) Text Transcription: \frac{x-1}{2}=\frac{y+2}{3}=\frac{2 z-1}{4} \frac{x+2}{-1}=\frac{2-y}{2} z=\frac{1}{2}

The x axis and r = j k + (2i 3j + k)t.

A particle is traveling along the line (x 3)/2=(y + 1)/(2) = z 1. Write the equation of its path in the form r = r0 + At. Find the distance of closest approach of the particle to the origin (that is, the distance from the origin to the line). If t represents time, show that the time of closest approach is t = (r0 A)/|A| 2. Use this value to check your answer for the distance of closest approach. Hint: See Figure 5.3. If P is the point of closest approach, what is A r?

We have seen that an orthogonal matrix with determinant 1 has at least one eigenvalue = 1, and an orthogonal matrix with determinant = 1 has at least one eigenvalue = 1. Show that the other two eigenvalues in both cases are ei, ei, which, of course, includes the real values 1 (when = 0), and 1 (when = ). Hint: See Problem 9, and remember that rotations and reflections do not change the length of vectors so eigenvalues must have absolute value = 1.

Find a unitary matrix U which diagonalizes A in (11.31) and verify that U1AU is diagonal with the eigenvalues down the main diagonal.

Show that an orthogonal matrix M with all real eigenvalues is symmetric. Hints: Method 1. When the eigenvalues are real, so are the eigenvectors, and the unitary matrix which diagonalizes M is orthogonal. Use (11.27). Method 2. From Problem 46, note that the only real eigenvalues of an orthogonal M are 1. Thus show that M = M1. Remember that M is orthogonal to show that M = MT.

Verify the results for F in the discussion of (11.34).

Show that the trace of a rotation matrix equals 2 cos + 1 where is the rotation angle, and the trace of a reflection matrix equals 2 cos 1. Hint: See equations (7.18) and (7.19), and Problem 10.

1 11 0 @ 269 6 7 6 9 6 2 1 A

1 2 0 @ 1 1 2 1 1 2 2 2 0 1 A

1 3 0 @ 122 2 1 2 2 2 1 1 A

1 2 0 @ 1 2 1 2 0 2 1 2 1 1 A

1 9 0 @ 184 4 4 7 8 1 4 1 A

1 2 2 0 @ 2 2 2 2 1+ 2 1 2 2 1 2 1+ 2 1 A

Show that if D is a diagonal matrix, then Dn is the diagonal matrix with elements equal to the nth power of the elements of D. Also show that if D = C1MC, then Dn = C1MnC, so Mn = CDnC1. Hint: For n = 2, (C1MC)2 = C1MCC1MC; what is CC1?

Note in Section 6 [see (6.15)] that, for the given matrix A, we found A2 = I, so it was easy to find all the powers of A. It is not usually this easy to find high powers of a matrix directly. Try it for the square matrix M in equation (11.1). Then use the method outlined in Problem 57 to find M4, M10, eM.

Repeat the last part of Problem 58 for the matrix M = 3 1 1 3 .

The Caley-Hamilton theorem states that A matrix satisfies its own characteristic equation. Verify this theorem for the matrix M in equation (11.1). Hint: Substitute the matrix M for in the characteristic equation (11.4) and verify that you have a correct matrix equation. Further hint: Dont do all the arithmetic. Use (11.36) to write the left side of your equation as C(D2 7D + 6)C1 and show that the parenthesis = 0. Remember that, by definition, the eigenvalues satisfy the characteristic equation.

At the end of Section 9 we proved that if H is a Hermitian matrix, then the matrix eiH is unitary. Give another proof by writing H = CDC1, remembering that now C is unitary and the eigenvalues in D are real. Show that eiD is unitary and that eiH is a product of three unitary matrices. See Problem 9.17d.

Show that if matrices F and G can be diagonalized by the same C matrix, then they commute. Hint: Do diagonal matrices commute?