Working with Vectors (AQA A-Level Mathematics): Revision Notes

• Notation: Vectors are often denoted by bold letters (e.g., A) or with an arrow above the letter (e.g., $\vec{A}$).
• Components: A vector in 2D space can be broken down into its components along the $x$ and $y$axes. For a vector $\vec{A}$, if $\vec{A}$ has a magnitude $A$ and an angle $\theta$ with the horizontal axis:

$\vec{A} = A_x \hat{i} + A_y \hat{j}$

where:

$A_x = A \cos \theta, \quad A_y = A \sin \theta$

$A_x$ and $A_y$ are the components of the vector along the $x$-axis and $y$-axis, respectively, and $\hat{i}$ and $\hat{j}$ are the unit vectors in the $x$ and $y$ directions.

• Graphical Method (Head-to-Tail Rule):
• To add two vectors $\vec{A}$ and $\vec{B}$, place the tail of $\vec{B}$ at the head of $\vec{A}$. The resultant vector $\vec{R}$ is drawn from the tail of $\vec{A}$ to the head of $\vec{B}$.
• Vector subtraction can be viewed as adding a negative vector. To subtract $\vec{B}$ from $\vec{A}$, reverse the direction of $\vec{B}$ and then add it to $\vec{A}$.
• Component Method:
• Add the corresponding components of each vector:

$\vec{R} = \vec{A} + \vec{B} = (A_x + B_x) \hat{i} + (A_y + B_y) \hat{j}$

• Similarly, for subtraction:

$\vec{R} = \vec{A} - \vec{B} = (A_x - B_x) \hat{i} + (A_y - B_y) \hat{j}$

• Magnitude:
• The magnitude (or length) of a vector $\vec{A}$ with components $A_x$ and $A_y$ is given by:

$|\vec{A}| = \sqrt{A_x^2 + A_y^2}$

• Direction:
• The direction of $\vec{A}$ can be found using the inverse tangent function:

$\theta = \tan^{-1} \left(\frac{A_y}{A_x}\right)$

• This angle $\theta$ is measured counter clockwise from the positive $x$-axis.

• Definition: The scalar product of two vectors $\vec{A}$ and $\vec{B}$ is a scalar quantity and is calculated as:

$\vec{A} \cdot \vec{B} = |\vec{A}| |\vec{B}| \cos \theta$

where $\theta$ is the angle between the two vectors.

• Component Form:
• In terms of components:

$\vec{A} \cdot \vec{B} = A_x B_x + A_y B_y$

• The scalar product is useful for finding the work done by a force: $W = \vec{F} \cdot \vec{d}$.

• Definition: The vector product of two vectors $\vec{A}$ and $\vec{B}$ is a vector that is perpendicular to the plane containing $\vec{A}$ and $\vec{B}$. Its magnitude is given by:

$|\vec{A} \times \vec{B}| = |\vec{A}| |\vec{B}| \sin \theta$

where $\theta$ is the angle between the two vectors.

• Direction: The direction of the vector product is given by the right-hand rule.
• Component Form:
• In terms of components for vectors in 3D:

$\vec{A} \times \vec{B} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ A_x & A_y & A_z \\ B_x & B_y & B_z \end{vmatrix}$

• The vector product is useful in torque calculations: $\vec{\tau} = \vec{r} \times \vec{F}$ .

• Condition for Equilibrium:
• An object is in equilibrium if the resultant force acting on it is zero. For a set of forces $\vec{F_1}, \vec{F_2}, \ldots, \vec{F_n}$ acting on a body, the condition for equilibrium is:

$\sum \vec{F} = \vec{F_1} + \vec{F_2} + \cdots + \vec{F_n} = \vec{0}$

• This requires that both the sum of the $x$-components and the sum of the $y$-components of the forces are zero.

• Resultant Force:
• Use vector addition to find the resultant force acting on an object when multiple forces are involved.
• Projectile Motion:
• Break down the velocity into horizontal and vertical components to analyse the motion separately along each axis.
• Relative Velocity:
• Calculate the velocity of one object relative to another using vector subtraction.

Working with vectors in A-Level Mechanics involves understanding how to represent, add, subtract, and multiply vectors. Mastery of these skills allows you to solve a wide range of problems involving forces, motion, and equilibrium, where both magnitude and direction are critical.