Some 3-D vector questions

ENGR 231 Winter 2015-16 1

ENGR 231 Homework 7 Due: Saturday 3/12 on the course web site.

Problem 1. A parallelogram is specified by vertices 0, a, b, and a+b in R2 has area A = a b sinθ .

(a) Show that

A 2 = a•a( ) b•b( )− a•b( )2

(b) Let D be the matrix whose columns are a and b. Show that

A 2 = det DTD( )

(c) Does the formula in part (b) give the correct value of area if the parallelogram is defined by vectors 0, a, b, and a+b are in R3?

Problem 2. Given a line L defined by x = a + αv, where а is an arbitrary real nuber. Let p be the point on the line closest to the origin.

(a) Show that

p= a− v

a•v v•v

Hint: p is orthogonal to Span v. (b) Let a = (5, 12) and v = (-1, 2). Find the point on the line closest to the origin

and the distance between the line and the origin. Problem 3.

A= 3 4 4 1 −4 −4

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⎥ ⎥ ⎥ , b=

1.7 6.8 −3.5

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⎥ ⎥ ⎥

a) Find x such that Ax−b is as small as possible. Hint: Normal Equations.

b) Let be = Ax, where x is found in a). Find be. c) Find ||b– be||. Verify that ||b||2 = ||b– be ||2 + || be ||2

Problem 4. We are given the following data: X -8 -4 2 6 8 Y 6.5 1.7 0.3 3.7 6.3

Fit the data with the formula y = ax2. (a) Construct the design matrix. (b) Find the normal equation. (c) Find the least squares value of a.

a

b

0

a+b

θ

L

av

ENGR 231 Winter 2015-16 2

Problem 5. We are given the following homogeneous differential equation: ′′x +3 ′x +2x =0

(a) Assume that x(t) = Ceλt. Find values of λ for which this satisfies the equation with C ≠ 0.

(b) Write a general solution for this homogeneous equation. (c) Is the solution stable or unstable? (d) How many initial values must be specified to obtain a unique solution?

Problem 6. We are given the following differential equation

(a) Find the equilibrium solution. (b) Write the homogenous differential equation for this problem. (c) Find values of λ for which the homogenous equation has a nonzero solution

of the form x = ceλt. (d) Is this system stable on unstable?

Problem 7. For each of the following problems: (a) Find the equilibrium solution, if it exists. (b) Find the characteristic equation and values of λ for which the homogeneous solution of the type x = ceλt can be nonzero, (c) State whether the system is stable or unstable.

a. b.

c. d.

′x + 0 −1 8 2

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⎦ ⎥x =

2 0

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⎦ ⎥

′x + 1 −1 2 4

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⎦ ⎥x =

3 0

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⎦ ⎥ ′x +

−1 1 −2 4

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⎦ ⎥x =

3 0

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⎦ ⎥

′x + −1 −5 −10 −1

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⎦ ⎥x =

4 11

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⎦ ⎥ ′x +

−4 2 2 −1

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⎦ ⎥x =

2 0

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⎦ ⎥