Specialist Maths - Unit 1 Topic 4: Algebra of vectors in 2D

Vectors

• Mathematical tool for dealing with motion and forces.

• Applications: displacement, velocity, forces in equilibrium, relative velocity.

• Isaac Newton's laws describe motion; vectors explain lunar and planetary motion.

Learning Intentions

• Multiply a vector by a scalar in Cartesian form.

• Determine a vector between two points.

• Determine if two vectors are parallel or perpendicular.

Position Vectors

• Vector between two points.

• Relative to the origin, position vector from O to A is $\overrightarrow{OA}$.

• Position vector from O to B is $\overrightarrow{OB}$.

• Position vector $\overrightarrow{AB} = \overrightarrow{OB} - \overrightarrow{OA}$.

• Position vector of $\overrightarrow{BA} = \overrightarrow{OA} - \overrightarrow{OB}$.

Equality of Two Vectors

• Two vectors are equal if their x & y components are the same.

• Vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ are only equal ($\overrightarrow{a} = \overrightarrow{b}$) if and only if $\overrightarrow{a} = \overrightarrow{b}$.

Scalar Multiplication of Vectors

• Multiply each component by the scalar: If vector $\overrightarrow{a} = x\hat{i} + y\hat{j}$ multiplied by scalar $k$, then $k\overrightarrow{a} = kx\hat{i} + ky\hat{j}$.

• Two vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ are parallel if $\overrightarrow{a} = k\overrightarrow{b}$, where $k ∈ R$.

  • If $\overrightarrow{a}$ is parallel to the y-axis, the i component = 0.

  • If $\overrightarrow{a}$ is parallel to the x-axis, the j component = 0.

Midpoint of a Line Segment

• The vector representing the midpoint, M, of the line segment AB can be found using: $\overrightarrow{OM} = \frac{1}{2}(\overrightarrow{OA} + \overrightarrow{OB})$ OR $\overrightarrow{OM} = \overrightarrow{OA} + \frac{1}{2}\overrightarrow{AB}$.

Scalar (Dot) Product

• The scalar or dot product of two vectors, $\overrightarrow{a}$ and $\overrightarrow{b}$, is denoted by $\overrightarrow{a} \cdot \overrightarrow{b} = |a||b|\cos(\theta)$.

• Properties:

Scalar Products in Component Form

• If $\overrightarrow{a} = a<em>1\hat{i} + a</em>2\hat{j}$ and $\overrightarrow{b} = b<em>1\hat{i} + b</em>2\hat{j}$, then $\overrightarrow{a} \cdot \overrightarrow{b} = a<em>1b</em>1 + a<em>2b</em>2$.

Finding the Angle Between Two Vectors

• Combining two methods: $\cos(\theta) = \frac{\overrightarrow{a} \cdot \overrightarrow{b}}{|a||b|}$.

• $\theta$ is between 0° and 180°.

Perpendicular Vectors

• If two vectors are perpendicular, the angle between them is 90°.

• Dot product: $\overrightarrow{a} \cdot \overrightarrow{b} = 0$.

Parallel Vectors

• If vector $\overrightarrow{a}$ is parallel to vector $\overrightarrow{b}$, then $\overrightarrow{a} = k\overrightarrow{b}$, where $k ∈ R$.

• The angle between them is 0° or 180°.

Projection of Vectors

• Scalar projection of $\overrightarrow{a}$ on $\overrightarrow{b}$: $|a|\cos(\theta) = \frac{\overrightarrow{a} \cdot \overrightarrow{b}}{|b|}$.

Vector Resolute

• Vector is the sum of two components:

Vector Operations

• Applications in surveying, navigation, orienteering.

• Vector addition and subtraction find resultant displacement and velocity vectors.

Displacement and Velocity

Force

• Equilibrium: When all forces acting on an object are balanced, the net or resultant force is zero.

Assumptions in Newtonian Dynamics

• Light (body or string): Object has no mass.

• Smooth: No frictional forces are exerted.

• Inextensible: Strings or ropes do not stretch.

• Rigid: Objects do not change shape when forces are applied.

• Perfectly elastic: Applied forces do not permanently deform an object.

Force Diagrams

• If an object is not moving or moving at a constant velocity, acceleration = 0, and the net force is 0.

Triangle of Forces

• If three non-parallel forces have a resultant force of zero, they can be represented in a triangle.

Newton’s First Law of Motion

• An object at rest remains at rest, and an object in motion remains in motion at constant speed unless acted on by an unbalanced force.

Mass and Weight

• Mass is a scalar, measured in kilograms.

• Weight is a vector quantity: $W = mg$ where g = 9.8 m/s².

Resolving Forces into Components

Forces and Equilibrium

• When FR=0, the body is in equilibrium and moves at a constant velocity.

Friction

• Frictional forces oppose motion.

Coefficient of Friction

• $\mu$ is a constant that varies for different surfaces.

Relative Velocity

• The relative velocity of object A with respect to object B is the velocity A would appear to have to an observer moving with B.