Some questions to practice your fundamental skills in vectors.

Some basic tasks involving vectors, including converting to/from component form, scalar product, resultant vectors.

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

Find eigenvalues and eigenvectors of a $2 \times 2$ matrix.

5 questions on vectors. Scalar product, angle between vectors, cross product, when are vectors perpendicular, combinations of vectors defined or not.

Some basic tasks involving vectors, including converting to/from component form, scalar product, resultant vectors.

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

Let $P_n$ denote the vector space over the reals of polynomials $p(x)$ of degree $n$  with coefficients in the real numbers. Let the linear map $\phi: P_4 \rightarrow P_4$ be defined by: \[\phi(p(x))=p(a)+p(bx+c).\]Using the standard basis for range and domain find the matrix given by $\phi$.

Let $P_n$ denote the vector space over the reals of polynomials $p(x)$ of degree $n$  with coefficients in the real numbers.

Let the linear map $\phi: P_4 \rightarrow P_4$ be defined by:

$\phi(p(x))=ap(x) + (bx + c)p'(x) + (x ^ 2 + dx + f)p''(x)$

Using the standard basis for range and domain find the matrix given by $\phi$.

Given $6$ vectors in $\mathbb{R^4}$ and given that they span $\mathbb{R^4}$ find a  basis.

Given three vectors, arrange them in a tip to tail arrangement using geogebra, then estimate the magnitude and direction of their resultant.

This question concerns the evaluation of the eigenvalues and corresponding eigenvectors of a 2x2 matrix.

Add three vectors by determining their scalar components, summing them and then resolving the rectangular components to find the magnitude and direction of the resultant

Add three vectors by determining their scalar components, summing them and then resolving the rectangular components to find the magnitude and direction of the resultant.

Given three vectors with integer components, find the corresponding magnitude and direction.

Given three vectors, arrange them in a tip to tail arrangement using geogebra, then estimate the magnitude and direction of their resultant.

Determine the resultant of three random 2-D vectors.  

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

Add three vectors by determining their scalar components and summing them.

Given a 3 x 3 matrix, and two eigenvectors find their corresponding eigenvalues. Also fnd the characteristic polynomial and using this find the third eigenvalue and a normalised eigenvector $(x=1)$.

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.

Find a unit vector orthogonal to two others.

Uses $\wedge$ for the cross product. The interim calculations should all be displayed to enough dps, not 3,  to ensure accuracy to 3 dps. If the cross product has a negative x component then it is not explained that the negative of the cross product is taken for the unit vector.

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

Given vectors $\boldsymbol{a,\;b}$, find $\boldsymbol{a\times b}$

rebelmaths

Find the dot product and the angle between two vectors

angle between two vectors

rebelmaths

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

Elementary operations on vectors; sum, modulus, unit vector, scalar multiple. 

Two column vectors are given, with one entry unknown. Given that the vectors are perpendicular, question is to determine the unknown entry.

Curl of a vector field.