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Scalar and Vector Operations

In physics problems, it is important to recognize when working with scalar or vector operations, as this can dramatically change the results. Scalar quantities are given by a data point, such as temperature, whereas vector quantities are given in a three-dimensional space, such as velocity. Preforming mathematical operations to a scalar quantity is simple. Just carry out the operation as normal and carry and convert the units. A vector quantity requires the direction of the vector to be taken into consideration.

Scalar quantities only deal with the magnitude of the medium. Scalar quantities is generally what is used. Time, volume, speed and temperature are common scalar quantities. They are just a data point with no respect to any direction. Vector quantities are what indicate direction and can become more complex in the mathematical computation. Increase/Decrease in temperature and velocity are common examples of vector quantities. It is possible to multiply a scalar and vector quantity together. The resulting quantity will be a vector.

net velocity due to water current and ship full power

A ship has a top speed of 3.0 m/s in calm water. The current of the ocean tends to push the boat at 2.0 m/s on a bearing of due south. What will be the net velocity of the ship if the captain points his ship on a bearing of 55° North of West and applies full power?

Scalar Potential

Please see the attached file for the adequate depiction of the scalar problem. A conservative force F can be written as F = -[upside down triangle]*phi symbol, where the phi symbol is called a scalar potential. Calculate the scalar potential for the gravitational force F = -k*abs. r/r^(2).

Determining the Center of a Mass

A square plate of uniform sheet metal, with edges of length 2a, is placed in the first quadrant of the xy plane with a corner at the origin. A circular hole, of diameter a, is centered in the quarter of the plate farthest from the origin, is removed from the plate. Where is the center of mass of the remaining metal?

Simple Cubic "Non-Bravais" Lattice

A simple cubic structure is constructed in which two planes of A atoms followed by two planes of B atoms alternate in the [100] direction (see attached). a) What is the crystal structure (viewed as a non-Bravais lattice with four atoms per unit cell)? b) What are the primitive translation vectors of the reciprocal lattice?

Use Cartesian tensor methods to prove the given entity.

Please show complete steps. 4. (a) If r denotes the position vector whose ith component is xi and b is a constant vector whose ith component is bi; use Cartesian tensor methods to prove that {Please refer to the attachment}. (b) The mutual potential energy associated with two dipoles of moments vector_b and vector_c is

Finding a vector's magnitude and direction

Consider four vectors F1, F2, F3, and F4 with magnitudes F1= 2 N, F2 = 24 N, F3 = 21 N, and F4 = 66 N, angle 1 = 150 degrees, angle 2 = -150 degrees, angle 3 = 32 degrees, angle 4 = -70 degrees, measured from the positive x axis with counterclockwise positive. 1) What is the magnitude of the resultant vector F = F1 + F2 + F

Directions and Magnitudes of Resultant Vectors

Three forces act through the point O in the directions shown in the figure in the attachment. The magnitudes of the forces are 15kN, 10kN and P. The resultant R acts at an angle of 10 degrees to Ox. Find the magnitudes of P and R.

Vector addition: Hiker's displacement

Path A is 9.0 km long heading 60.0° north of east. Path B is 6.0 km long in a direction due east. Path C is 3.0 km long heading 315° counterclockwise from east. (a) Graphically add the hiker's displacements in the order A, B, C. What is the magnitude of displacement in km? Direction of displacement? (b) Graphically add t

Vector Common Point Angles

When two vectors A and B are drawn from a common point, the angle between them is (phi) . If A and B have the same magnitude, for which value of (phi) will their vector sum have the same magnitude as A or B? --------------------------------------------------------------------------------------------------- Three hori

Hilbert Space

1. What constitutes a Hilbert Space and how is it different (if it is) from the usual vector space? 2. What does it mean to "normalize" a function in QM vs normalize a vector in standard vector notation. Don't be afraid to be basic or elementary!

Mass in metric, mass vs weight, law of motion, net force, vector

1. Answer the following: a. What are the units of mass and the units of weight for the metric system? b. What are the units of mass and the units of weight for the English system? 2. Explain the relationship between the mass and the weight of an object. Is the mass of an object the same on earth and the moon? Are the wei

Vector Problems: Write out the Equations.

1. Write out the equation for Vector Ax. Let A's angle wrt +x axis be θ. wrt means With Respect To. 2. Similar to the previous question, what is the equation for Vector Ay. 3. As in the previous two questions, write out the equation for the magnitude of A ( |A| = ? ) using A's x and y components.

Vector A and B problems

1. Vector A = 119 grams @ 164°. What is Ax ? 2. Vector A = 119 grams @ 164°.What is Ay ? 3. Vector B = (114) xunit vector + (-32.8) yunit vector. Assume [grams] as the unit. What is the magnitude of B? 4. Vector B = (114) xunit vector + (-32.8) yunit vector. Assume [grams] as the unit. What is the direction of B in ra

Orthonormal Basis Vectors

Given two sets of complete orthonormal basis vectors: {|u1>, |u2>, |u3>,...} and {|v1>, |v2>, |v3>,...}. Use Dirac notation to prove: 1. That for any linear operator Â, the trace is independent of choice of basis i.e. Tr(Â) = Σ<ui| Â|ui> = Σ<vi| Â|vi>. 2. Given that Tr(Â) = Σ<ui| Â|ui>. Prove that Tr(ÂĈ) =

Normalized kets

The state space of a certain physical system is 3-D. Let { |u1 , | u2 , | u3 } be an orthonormal basis of this space. The kets | phi0 ? and | phi1 ? are defined by: | phio = (1/sqrt2 ) | u1 + (i/2) | u2 + (1/2) | u3 | phi1 = (1/sqrt3 ) | u1 + (i/sqrt3) | u3 1.Are these kets normalized? 2.Calculate the matrices

The inertia tensor and its principal axes

A three particles system consists of masses mi and coordinates (x1,x2,x3) as follows (see file for complete details). Calculate the tensor of inertia, the principal axes and principal moments of inertia.

Cubic and Orthorhombic Crystals

Show that the general direction [ hkl ] in a cubic crystal is normal to the planes with Miller indices (hkl). Is the same true in general for an orthorhombic crystal? Show that the spacing d of the (hkl) set of planes in a cubic crystal with lattice parameter a is: d = (a)/(h^2 + k^2 +l^2)^(1/2) What is the generaliza

Eigenvalues and Eigenvectors of a Hermitian Matrix

Please see the attached file for the full problem statement, and please show all steps in your solution. Consider the Hermitian matrix omega = 1/2 [2 0 0 0 3 -1 0 -1 3] (1) Show that omega1 = omega2 = 1; omega3 = 2. (2) Show that |omega = 2> is any vector of the form 1/((2a^2)^(1/2)) [0 a -a] (3) Show that t

Physics vectors and distance

Vector b has magnitude 7.1 and direction 14 degrees below the +x-axis. Vector c has x-component cx = -1.8 and y-component cy=6.7. Compute: A) the x and y components of b; B) the magnitude and directions of c; C) the magnitude and direction of c + b(resultant)?

Work, vectors, and kinetic energy

4. A 1.5 kg object moving along the x axis has a velocity of +4.0 m/s at x = 0. If the only force acting on this object is shown in the figure, what is the kinetic energy of the object at x = +3.0m? 5. If the vectors A and B have magnitudes of 10 and 11, respectively, and the scalar product of these two vectors is -100, what

displacement vector and distance

Hello I need help answering these problems. Peter noticed a bug crawling along a meter stick and decided to record the bug's position in five-second intervals. After the bug crawled off the meter stick, peter created the table shown. 1. What is the displacement of the bug between t = 0.00 s and t = 20.0 s? 2. what is the to

Curves in euclidean 3-space

In Euclidean three-space, let p be the point with coordinates (x,y,z) = (1,0,-1). Consider the following curves that pass through p: Curve 1: xi (λ) = (λ, (λ-1)2, - λ) Curve 2: xi (μ) = (cos μ , sin μ, μ-1) Curve 3: xi (σ) = (σ2, σ3 + σ2, σ) The curves are parametrized by the parameters that v

Gradient and divergence formulas

See attached file for full problem description. Derive the gradient formula for a potential in both cylindrical and spherical coordinates. Also, derive the divergence formula of the vector field in both coordinates.

Applications of the Stokes Theorem

Stokes Theorem. See attached file for full problem description. 1. compute the line integral where F = (yz^2 - y)i + (xz^2 + x)j + 2xyzk where C is the circle of radius 3 in the xy-plane, centered at the origin, oriented counterclockwise as viewed from the positive z -axis. 2. Given F =yi - xj + yzk and the region S determ

Moment of inertia of a system of masses.

An object is made up of three masses connected by massless rods of fixed length. Mass A is located at (30.0 cm, 0 cm) and has a mass of 250 grams, mass B is located at (0 cm, 30.0 cm) and has a mass of 350 grams, mass C is located at (-30.0 cm, 0 cm) and has a mass of 450 grams. What is the moment of inertia of this object about

Eigenvalues, eigenvectors, and time evolution.

Dear Mitra, I in fact wrote the wrong matrix but I am still confused after PART A. (PART A) The actual matrix is: H = 1 2 0 2 0 2 0 2 -1 Where the eigenvalues are E1 =0, E2=3hw, E3=-3hw and as you said the trace(H) =0 = sum of eigenvalues. I also found the eigenvectors using Hx = Ex and they w

Problems involving scalar and vector operations

The eastward component of vector A is equal to the westward component of vector B and their northward components are equal. Which one of the following statements is correct for these two vectors? Choices: Vector A is parallel to vector B Vector A is anti-parallel to vector B The magnitude of vector A is equal to the mag