Homework Answers
1) cross product is taken when both the quantities are vector.therefore when 2 vector makes angle with each other we need to take theta as an angle between them,for there cross product.
2) (a) when line of action of force makes 0 degree or 180 degree with the radius vector.
(b) when either the force is zero or perpendicular distance is zero.
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Cross Product Extra Credit Opportunity: Explain why the theta in cross product is defined as the...
8. ExTRA CREDIT. Let Vz be the branch of the square root function defined in Question 3 of Projec...
8. ExTRA CREDIT. Let Vz be the branch of the square root function defined in Question 3 of Project 4, with domain the entire complex plane C. Find a subdomain Ω of C such that the formula V212VVz2 would hold for any z1,z2 S2. Show why the formula tails in all of C, and provide concrete examples. Generalize your answer for the case when the branch of chosen so that Carg%, α + 2π, where α is any fixed real...
Extra Credit: For the following two extra credit problems, look at the file "Notes on inner...
Extra Credit: For the following two extra credit problems, look at the file "Notes on inner product spaces." Which will be posted on Blackboard on Tuesday 6/25 10. Consider the space P (degree 2 polynomials) with inner product f. 9) f()g(t)dt. Extra Credit: For the following two extra credit problems, look at the file "Notes on inner product spaces." Which will be posted on Blackboard on Tuesday 6/25 10. Consider the space P2 (degree 2 polynomials) with inner product S(t)g(t)dt....
Extra credit problem (5 pt) The isothermal bulk modulus (Br) is defined as the reciprocal of...
Extra credit problem (5 pt) The isothermal bulk modulus (Br) is defined as the reciprocal of the isothermal compressibility (KIn p/aP)r). Show that the modulus is related to the chemical potential by where p is the number density
Defining the cross product The cross product of two nonzero vectors
Defining the cross product The cross product of two nonzero vectors \(\vec{u}\) and \(\vec{v}\) is another vector \(\vec{u} \times \vec{v}\) with magnitude$$ |\vec{u} \times \vec{v}|=|\vec{u}||\vec{v}| \sin (\theta), $$where \(0 \leq \theta \leq \pi\) is the angle between the two vectors. The direction of \(\vec{u} \times \vec{v}\) is given by the right hand rule: when you put the vectors tail to tail and let the fingers of your right hand curl from \(\vec{u}\) to \(\vec{v}\) the direction of \(\vec{u} \times \vec{v}\)...
5. (a) Explain why the standard inner product is invariant under an orthogonal trans formation. T...
5. (a) Explain why the standard inner product is invariant under an orthogonal trans formation. That is, if U is any orthogonal miatrix, and if u = Ux and v = Uy, then i.e. multiplication by an orthogonal matrix does not change the standard inner product. (b) Given any two vectors x. y in R", explain why the angle between them is Py invarient under an orthogonal transformation. That is, if u where P is an orthogonal matrix, thern Px...
1) Use Snell's Law (na sin(theta)a = nb sin(theta)b) to explain why (theta)b is greater than...
1) Use Snell's Law (na sin(theta)a = nb sin(theta)b) to explain
why (theta)b is greater than (theta)a in Fig.2
2) In Fig. 2, if we increase the index nb but keep nb < na,
should the value of (theta)c increase or decrease? Give your
reasoning.
3) Does total internal reflection occur if the refraction of
light is passing from ait into water? Why?
1 b 2 Өь Нө = 90° пь 3 na Өa ec > Ө 4 a Р...
Extra Credit (+3% maximum): The range R of a projectile fired with an initial velocity vo...
Extra Credit (+3% maximum): The range R of a projectile fired with an initial velocity vo at an angle with the horizontal is given by R = ama where g is the acceleration due to gravity. Find the acute angle such that the range is a maximum.
name a product that today is not a gender cross over, but could be and explain...
name a product that today is not a gender cross over, but could be and explain why you selected this product.
Implicit costs are defined by economists as nonmonetary opportunity costs. Why is it important for a...
Implicit costs are defined by economists as nonmonetary opportunity costs. Why is it important for a firm to take these costs into consideration when evaluating a potential activity, when they don’t involve any monetary expense?
Please explain why x = x + 1 is Theta(nlgn) code: i = n; while (i...
Please explain why x = x + 1 is Theta(nlgn) code: i = n; while (i > 1) { i = floor(i/2); x = x + 1; } why is is theta(nlgn)?
8. ExTRA CREDIT. Let Vz be the branch of the square root function defined in Question 3 of Project 4, with domain the entire complex plane C. Find a subdomain Ω of C such that the formula V212VVz2 would hold for any z1,z2 S2. Show why the formula tails in all of C, and provide concrete examples. Generalize your answer for the case when the branch of chosen so that Carg%, α + 2π, where α is any fixed real...
Extra Credit: For the following two extra credit problems, look at the file "Notes on inner product spaces." Which will be posted on Blackboard on Tuesday 6/25 10. Consider the space P (degree 2 polynomials) with inner product f. 9) f()g(t)dt. Extra Credit: For the following two extra credit problems, look at the file "Notes on inner product spaces." Which will be posted on Blackboard on Tuesday 6/25 10. Consider the space P2 (degree 2 polynomials) with inner product S(t)g(t)dt....
Extra credit problem (5 pt) The isothermal bulk modulus (Br) is defined as the reciprocal of the isothermal compressibility (KIn p/aP)r). Show that the modulus is related to the chemical potential by where p is the number density
Defining the cross product The cross product of two nonzero vectors \(\vec{u}\) and \(\vec{v}\) is another vector \(\vec{u} \times \vec{v}\) with magnitude$$ |\vec{u} \times \vec{v}|=|\vec{u}||\vec{v}| \sin (\theta), $$where \(0 \leq \theta \leq \pi\) is the angle between the two vectors. The direction of \(\vec{u} \times \vec{v}\) is given by the right hand rule: when you put the vectors tail to tail and let the fingers of your right hand curl from \(\vec{u}\) to \(\vec{v}\) the direction of \(\vec{u} \times \vec{v}\)...
5. (a) Explain why the standard inner product is invariant under an orthogonal trans formation. That is, if U is any orthogonal miatrix, and if u = Ux and v = Uy, then i.e. multiplication by an orthogonal matrix does not change the standard inner product. (b) Given any two vectors x. y in R", explain why the angle between them is Py invarient under an orthogonal transformation. That is, if u where P is an orthogonal matrix, thern Px...
1) Use Snell's Law (na sin(theta)a = nb sin(theta)b) to explain
why (theta)b is greater than (theta)a in Fig.2
2) In Fig. 2, if we increase the index nb but keep nb < na,
should the value of (theta)c increase or decrease? Give your
reasoning.
3) Does total internal reflection occur if the refraction of
light is passing from ait into water? Why?
1 b 2 Өь Нө = 90° пь 3 na Өa ec > Ө 4 a Р...
Extra Credit (+3% maximum): The range R of a projectile fired with an initial velocity vo at an angle with the horizontal is given by R = ama where g is the acceleration due to gravity. Find the acute angle such that the range is a maximum.
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