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

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.

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.

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

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

Determine if various combinations of vectors are defined or not.

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.

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

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

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.

Calculation of the length and alternative form of the parameteric representation of a curve.

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

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

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

Three 3 dim vectors, one with a parameter $\lambda$ in the third coordinate. Find value of $\lambda$ ensuring vectors coplanar. Scalar triple product.

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

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

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.

Given two 3 dim vectors, find vector equation of line through one vector in the direction of another. Find two such lines and their point of intersection.

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

Statistics and probability. Two questions: using LSD and Tukey yardsticks.

Practice bank of multivariable calculus questions

Gradient of $f(x,y,z)$.

Should include a warning to insert * between multiplied terms

Find the coordinates of the stationary point for $f: D \rightarrow \mathbb{R}$: $f(x,y) = a + be^{-(x-c)^2-(y-d)^2}$, $D$ is a disk centre $(c,d)$.

Find the stationary points of the function: $f(x,y)=a x ^ 3 + b x ^ 2 y + c y ^ 2 x + dy$ by choosing from a list of points.

Inputting the values given into the partial derivatives to see if 0 is obtained is tedious! Could ask for the factorisation of equation 1 as the solution uses this. However there is a problem in asking for the input of the stationary points - order of input and also giving that there is two stationary points.

Two double integrals with numerical limits

Calculate a repeated integral of the form $\displaystyle I=\int_0^1\;dx\;\int_0^{x^{m-1}}mf(x^m+a)dy$

The $y$ integral is trivial, and the $x$ integral is of the form $g'(x)f'(g(x))$, so it straightforwardly integrates to $f(g(x))$.

3 Repeated integrals of the form $\int_a^b\;dx\;\int_c^{f(x)}g(x,y)\;dy$ where $g(x,y)$ is a polynomial in $x,\;y$ and $f(x)$ is a degree 0, 1 or 2 polynomial in $x$.

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

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

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.