Divergence of vector fields.

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Find the dot product and the angle between two vectors

Find the cosine of the angle between two pairs of 3D and 4D vectors.

Some questions to practice your fundamental skills in vectors.

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 $(x=1)$.

$A,\;B$ $2 \times 2$ matrices. Find eigenvalues and eigenvectors of both. Hence or otherwise, find $B^n$ for largish $n$.

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

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

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

Determine if various combinations of vectors are defined or not.

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

Find the unit vector parallel to a given vector.

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

Divergence of vector fields.

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angle between two vectors

rebelmaths

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

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

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

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

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

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

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

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

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

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Using Jsxgraph to draw the vector field for a differential equation of the form $\frac{dy}{dx}=f(x,y)=x^2-y^2$, and also by moving the point $(x_0,y_0)$ you can see the solution curves going through that point.

If you want to modify $f(x,y)$ simply change the definition of  $f(x,y)$ and that of the variable  str in the user defined function testfield in Extensions and scripts. You have to use javascript notation for functions and powers in the definition of $f(x,y)$.