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

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. 

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angle between two vectors

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

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

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

This exercise will help you solve questions based on subduction zones using  the Cosine rule.

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

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