Sixth Term Examination Paper (STEP) Pure Mathematics: Vectors and Matrices Syllabus

Pure Mathematics: Vectors and Matrices syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Pure Mathematics: Vectors and Matrices in Sixth Term Examination Paper (STEP), not a summary of it.

1. 1

   Vectors and Geometry

   6 topics

   • Vectors in two and three dimensions; magnitude and unit vectors
   • The scalar (dot) product and angle between vectors
   • The vector (cross) product and its geometric meaning
     • Area and volume interpretations
   • Vector equations of lines and intersection/skew analysis
   • Vector and Cartesian equations of planes
     • Distance from a point to a line or plane
     • Angle between lines and planes
   • Scalar triple product and coplanarity
2. 2

   Matrices and Linear Transformations

   6 topics

   • Matrix algebra: addition, multiplication and properties
   • Determinants of 2x2 and 3x3 matrices
   • Inverse matrices and solving simultaneous equations
     • Geometric interpretation of singular systems
   • Matrices as linear transformations of the plane
     • Rotations, reflections, enlargements and shears
     • Composition of transformations
   • Invariant points and invariant lines
   • Eigenvalues, eigenvectors and diagonalisation
3. 3

   Further Algebraic Structures

   4 topics

   • Hyperbolic functions and their identities
     • Inverse hyperbolic functions in logarithmic form
     • Calculus of hyperbolic functions
   • Roots of unity and cyclotomic structure
   • Summation of series using complex numbers
   • Polar coordinates: curves, areas and tangents
     • Sketching polar curves
     • Area enclosed by polar curves

Pure Mathematics: Vectors and Matrices flashcards for Sixth Term Examination Paper (STEP)

22 of 50 cards from the Pure Mathematics: Vectors and Matrices deck — real questions with worked answers.

1. How is the magnitude (modulus) of a vector $\vec{a} = \begin{pmatrix} a_1 \\ a_2 \\ a_3 \end{pmatrix}$ computed?

   $|\vec{a}| = \sqrt{a_1^{2} + a_2^{2} + a_3^{2}}$. It is the Euclidean length of the vector.

2. What is a unit vector, and how do you obtain the unit vector $\hat{a}$ in the direction of a nonzero vector $\vec{a}$?

   A unit vector has magnitude $1$. The unit vector in the direction of $\vec{a}$ is $\hat{a} = \dfrac{\vec{a}}{|\vec{a}|}$.

3. Define the scalar (dot) product of $\vec{a}$ and $\vec{b}$ both algebraically and geometrically.

   Algebraically $\vec{a}\cdot\vec{b} = a_1 b_1 + a_2 b_2 + a_3 b_3$. Geometrically $\vec{a}\cdot\vec{b} = |\vec{a}||\vec{b}|\cos\theta$, where $\theta$ is the angle between them.

4. How do you find the angle $\theta$ between two nonzero vectors $\vec{a}$ and $\vec{b}$?

   $\cos\theta = \dfrac{\vec{a}\cdot\vec{b}}{|\vec{a}||\vec{b}|}$, so $\theta = \arccos\!\left(\dfrac{\vec{a}\cdot\vec{b}}{|\vec{a}||\vec{b}|}\right)$.

5. What does the dot product tell you about whether two nonzero vectors are perpendicular?

   Two nonzero vectors are perpendicular if and only if $\vec{a}\cdot\vec{b} = 0$ (since $\cos 90^{\circ}=0$).

6. State the formula for the scalar projection of $\vec{a}$ onto $\vec{b}$.

   The scalar projection (component of $\vec{a}$ along $\vec{b}$) is $\dfrac{\vec{a}\cdot\vec{b}}{|\vec{b}|} = \vec{a}\cdot\hat{b}$.

7. Define the vector (cross) product $\vec{a}\times\vec{b}$ using the determinant form.

   $\vec{a}\times\vec{b} = \begin{vmatrix} \vec{i} & \vec{j} & \vec{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix} = \begin{pmatrix} a_2 b_3 - a_3 b_2 \\ a_3 b_1 - a_1 b_3 \\ a_1 b_2 - a_2 b_1 \end{pmatrix}$.

8. What is the geometric meaning of the magnitude of the cross product $|\vec{a}\times\vec{b}|$?

   $|\vec{a}\times\vec{b}| = |\vec{a}||\vec{b}|\sin\theta$, which equals the area of the parallelogram spanned by $\vec{a}$ and $\vec{b}$. Half of this is the area of the triangle.

9. Describe the direction of $\vec{a}\times\vec{b}$.

   It is perpendicular to both $\vec{a}$ and $\vec{b}$, with orientation given by the right-hand rule. If $\vec{a}\parallel\vec{b}$ then $\vec{a}\times\vec{b}=\vec{0}$.

10. Is the cross product commutative? State the key algebraic property.

    No, it is anticommutative: $\vec{a}\times\vec{b} = -(\vec{b}\times\vec{a})$. Also $\vec{a}\times\vec{a}=\vec{0}$.

11. Write the vector equation of a line through point with position vector $\vec{a}$ and direction $\vec{d}$.

    $\vec{r} = \vec{a} + \lambda\vec{d}$, where $\lambda\in\mathbb{R}$ is a parameter.

12. How do you determine whether two lines $\vec{r}=\vec{a}+\lambda\vec{d}_1$ and $\vec{r}=\vec{b}+\mu\vec{d}_2$ intersect, are parallel, or are skew?

    Parallel if $\vec{d}_1\parallel\vec{d}_2$. Otherwise set the equations equal and solve component-wise: if a consistent $(\lambda,\mu)$ exists they intersect; if no solution exists the lines are skew.

13. What does it mean for two lines in three dimensions to be skew?

    Skew lines are not parallel and do not intersect; they lie in different planes and have no common point.

14. Give the scalar (normal) form of the vector equation of a plane with normal $\vec{n}$ passing through point $\vec{a}$.

    $\vec{r}\cdot\vec{n} = \vec{a}\cdot\vec{n}$, equivalently $(\vec{r}-\vec{a})\cdot\vec{n}=0$.

15. How do you convert the plane equation $\vec{r}\cdot\vec{n}=d$ with $\vec{n}=\begin{pmatrix} a\\ b\\ c\end{pmatrix}$ into Cartesian form?

    It becomes $ax + by + cz = d$, where $(a,b,c)$ are the components of the normal vector.

16. Write the parametric (two-direction) vector equation of a plane through $\vec{a}$ containing directions $\vec{u}$ and $\vec{v}$.

    $\vec{r} = \vec{a} + s\vec{u} + t\vec{v}$, with parameters $s,t\in\mathbb{R}$ and $\vec{u},\vec{v}$ non-parallel.

17. How can you find a normal vector to a plane given two non-parallel direction vectors $\vec{u}$ and $\vec{v}$ lying in it?

    Take the cross product: $\vec{n} = \vec{u}\times\vec{v}$.

18. State the formula for the scalar triple product of $\vec{a}$, $\vec{b}$, $\vec{c}$.

    $[\vec{a},\vec{b},\vec{c}] = \vec{a}\cdot(\vec{b}\times\vec{c}) = \begin{vmatrix} a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ c_1 & c_2 & c_3 \end{vmatrix}$.

19. What is the geometric meaning of the scalar triple product, and what does it being zero indicate?

    $|\vec{a}\cdot(\vec{b}\times\vec{c})|$ is the volume of the parallelepiped formed by the three vectors. If it equals $0$, the vectors are coplanar (linearly dependent).

20. Find the angle between a line with direction $\vec{d}$ and a plane with normal $\vec{n}$.

    $\sin\theta = \dfrac{|\vec{d}\cdot\vec{n}|}{|\vec{d}||\vec{n}|}$, where $\theta$ is the acute angle between the line and the plane.

21. State the conditions required for matrix addition and matrix multiplication to be defined.

    Addition: matrices must have the same dimensions. Multiplication $AB$: the number of columns of $A$ must equal the number of rows of $B$; if $A$ is $m\times n$ and $B$ is $n\times p$, then $AB$ is $m\times p$.

22. Is matrix multiplication commutative? State the relevant property and the transpose rule for products.

    No: in general $AB \neq BA$. Matrix multiplication is associative and distributive. The transpose of a product reverses order: $(AB)^{T} = B^{T}A^{T}$.

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