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 $(x=1)$.

Find the dot product and the angle between two vectors

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

rebelmaths

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

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

rebelmaths

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

rebelmaths

Questions on vector arithmetic and vector operations, including dot and cross product.

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

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

Find the dot product and the angle between two vectors

Putting a pair of linear equations into matrix notation and then solving by finding the inverse of the coefficient matrix. 

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

Practical questions

Find the dot product and the angle between two vectors

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

rebelmaths

Find the angle between two vectors

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

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

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

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

This is the question for week 9 of the MA100 course at the LSE. It looks at material from chapters 17 and 18.

Description of variables for part b:
For part b we want to have four functions such that the derivative of one of them, evaluated at 0, gives 0; but for the rest we do not get 0. We also want two of the ones that do not give 0, to be such that the derivative of their sum, evaluated at 0, gives 0; but when we do this for any other sum of two of our functions, we do not get 0. Ultimately this part of the question will show that even if two functions are not in a vector space (the space of functions with derivate equal to 0 when evaluated at 0), then their sum could nonetheless be in that vector space. We want variables which statisfy:

a,b,c,d,f,g,h,j,k,l,m,n are variables satisfying

Function 1: x^2 + ax + b sin(cx)
Function 2: x^2 + dx + f sin(gx)
Function 3: x^2 + hx + j sin(kx)
Function 4: x^2 + lx + m sin(nx)

u,v,w,r are variables satifying
u=a+bc
v=d+fg
w=h+jk
r=l+mn

The derivatives of each function, evaluated at zero, are:
Function 1: u
Function 2: v
Function 3: w
Function 4: r

So we will define
u as random(-5..5 except(0))
v as -u
w as 0
r as random(-5..5 except(0) except(u) except(-u))

Then the derivative of function 3, evaluated at 0, gives 0. The other functions give non-zero.
Also, the derivative of function 1 + function 2 gives 0. The other combinations of two functions give nonzero.

We now take b,c,f,g,j,k,m,n to be defined as \random(-3..3 except(0)).
We then define a,d,h,l to satisfy
u=a+bc
v=d+fg
w=h+jk
r=l+mn

Description for variables of part e:

Please look at the description of each variable for part e in the variables section, first.
As described, the vectors V3_1 , V3_2 , V3_3 are linearly independent. We will simply write v1 , v2 , v3 here.
In part e we ask the student to determine which of the following sets span, are linearly independent, are both, are neither:

both: v1,v2,v3
span: v1,v1+v2,v1+v2+v3, v1+v2+v3,2*v1+v2+v3
lin ind: v1+v2+v3
neither: v2+v3 , 2*v2 + 2*v3
neither:v1+v3,v1-2*v3,2*v1-v3
neither: v1+v2,v1-v2,v1-2*v2,2*v1-v2

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

Find the sum of two 2-dimensional vectors, graphically and exactly using the parallelogram rule.

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

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

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

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

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

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

Curl of a vector field.