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• Question

Curl and divergence of a vector field.  Determine whether the vector field is irrotational or solenoidal.

• Question

Multiplication and addition of complex numbers. Four parts.

• Question

Adding linear combinations of  polynomials over $\mathbb{Z}_3,\;\mathbb{Z}_5,\;\mathbb{Z}_7$

• Question

Correct an error in a received message which was encoded using Hamming's square code

• Hamming's [7,4] code
Question

Decode a message of two codewords encoded using Hamming's [7,4] code, with at most one error per codeword.

• Question

Compute the information rate and minimum distance of some binary codes.

• Question

Find upper and lower bounds on the number of codewords in three maximal codes given their codeword lengths and minimum distances.

Uses Hamming, Singleton and Gilbert-Varshamov bounds.

• Question

List all vectors in a spanning set.

(repeated 3 times)

• Question

Write down a lexicographic parity check matrix for a Hamming code and correct two received codewords.

• Question

Write down the lexicographic parity check matrix for a Hamming code, and correct two received codewords.

• Question

Compute the word length, minimum distance and dimension of some given Hamming codes.

• Question

Parametric form of a curve, cartesian points, tangent vector, and speed.

• Cartesian equation of a plane
Has some problems
Question

A plane goes through three given non-collinear points in 3-space. Find the Cartesian equation of the plane in the form $ax+by+cz=d$.

There is a problem in that this equation can be multiplied by a constant and be correct. Perhaps d can be given as this makes a,b and c unique and the method of the question will give the correct solution.

• Question

Calculation of the length and alternative form of the parameteric representation of a curve, involving trigonometric functions.

• Question

Find the Cartesian form $ax+by+cz=d$ of the equation of the plane $\boldsymbol{r=r_0+\lambda a+\mu b}$.

The solution is not unique. The constant on right hand side could be given to ensure that the left hand side is unique.

• Question

Determine if various combinations of vectors are defined or not.

• Question

When are vectors $\boldsymbol{v,\;w}$ orthogonal?

Part b) is not answered in Advice, the given solution is for a different question.

• Question

Find minimum distance between a point and a line in 3-space. The line goes through a given point in the direction of a given vector.

The correct solution is given, however the accuracy of 0.001  is not enough as in some cases answers near to the correct solution are also marked as correct.

• Angle between vectors
Question

Given vectors  $\boldsymbol{v,\;w}$, find the angle between them.

• Find the angle between planes
Has some problems
Question

Find angle between plane $\Pi_1$, given by three points, and the plane $\Pi_2$ given in Cartesian form.

The calculation of $cos(\alpha)$ at the end of Advice has fractionNumbers switched on and so the result is presented as a fraction, which can be misleading. Best if calculation is followed through without using fractionNumbers.

• Question

Calculations of the lengths of two 3D vectors, the distance between their terminal points, their sum, difference, and dot and cross products.

• Question

Given vectors $\boldsymbol{v}$ and $\boldsymbol{w}$, find their inner product.

• Question

(Green’s theorem). $\Gamma$ a rectangle, find: $\displaystyle \oint_{\Gamma} \left(ax^2-by \right)\;dx+\left(cy^2+px\right)\;dy$.

• Question

Find all words with given Hamming distance from a given codeword.

• Equivalent codes
Question

Compute tables of Hamming distances in given codes, then determine which codes are equivalent.

• Vector cross product
Question

Given vectors $\boldsymbol{A,\;B}$, find $\boldsymbol{A\times B}$

• Question

Given a set of vectors, find a basis which generates their span as a subspace of $\mathbb{Z}_n$.

• Question

Given a pair of 3D position vectors, find the vector equation of the line through both.  Find two such lines and their point of intersection.

• Vectors
Draft
Exam (11 questions)

Questions on vector arithmetic and vector operations, including dot and cross product, as well as the vector equations of planes and lines.

• Question

Given a matrix in the field $\mathbb{Z}_n$. By reducing it to row-echelon form (or otherwise), find a basis for the row space of the matrix, as a list of vectors.