Chapter 3 - Applications of Vectors

MATH1050Vectors

Now that we are familiar with the different representations of vectors, we can investigate some of the ways vectors are used to solve mathematical problems.

In pure mathematics, vectors can be used to solve problems in geometry.

In applied mathematics, vectors can be used to model quantities such as displacement, velocity, or forces acting on an object, and thus answer questions about such vector quantities.

There are two operations on vectors, the scalar product and the vector product, that also have useful applications.

In this section we look at several applications of vectors and investigate the scalar and vector products.

Topics in this section are:

• Vectors in Geometry
• Forces
• Displacement, velocity and momentum
• The scalar product
• The vector product - Part 1

3.1 Vectors in Geometry

Let $P$ be a point on the line through points $A$ and $B$. We can use vectors to describe the position vector of $P$ in terms of the position vectors of $A$ and $B$.

Let $\mathbf{p}=\overrightarrow{OP},\mathbf{a}=\overrightarrow{OA}\text{and}\mathbf{b}=\overrightarrow{OB}$.

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$$\begin{array}{c}\begin{array}{rl}& \text{Now}\mathbf{a}+\overrightarrow{AB}=\mathbf{b}\text{so}\overrightarrow{AB}=\mathbf{b}-\mathbf{a}.\text{Thus,}\\ & \mathbf{p}=\mathbf{a}+\overrightarrow{AP}=\mathbf{a}+t\overrightarrow{AB}=\mathbf{a}+t(\mathbf{b}-\mathbf{a})=(1-t)\mathbf{a}+t\mathbf{b},\\ & \text{where}t\text{is the real number such that}\overrightarrow{AP}=t\overrightarrow{AB}.\end{array}\end{array}$$

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3.2 Forces

A Force is an influence that can change the motion of an object and can be modelled as a vector.

Newton's Laws of Motion

Newton’s first law: the law of inertia: Newton’s first law states that if a body is at rest or moving at a constand speed in a straight line, it will remain at rest or keep mobing in a straight line at constant speed unless it is acted upon by a force.

Newton’s second law $\vec{\mathbf{F}}=m\vec{a}$

Newton’s third law: Law of action and reaction: To every action there is an opposed force acting against it.

The standard unit of measurement for forces is the newton, denoted $N$.

Weight

This is the force due to gravity, and we will denote it by $\mathbf{W}$. the weight vector associated with an object of mass $m$ kilograms has magnitude $9m$ newtons and has the direction vertically downwards.

Tension

Normal Reaction

Friction

This is a force in the direction parallel to a surface. It is the force that counteracts an object sliding along a surface. We will denote it as $\mathbf{f}$.

If exactly three forces are acting on an object, and that object is at rest, then the three vectors representing the forces must sum to zero. Geometrically, this means that the three vectors must form a triangle, with the arrows pointing in a consistent direction around the triangle.

3.3 Displacement, Velocity, and Momentum

Example 3.3.1

$\text{A surveyor walks 200 metres due North. He then turns clockwise through an angle of}\frac{2\pi }{3}\text{radians and walks 100 metres. Finally he turns and walks 300 metres due West.}\text{Find his resulting Displacement, relative to his starting point.}$

$$\begin{gathered} \begin{array}{c}\overrightarrow{OR}=\overrightarrow{OP}+\overrightarrow{PQ}+\overrightarrow{QR} \\ \overrightarrow{OP}=0\mathbf{i}+200\mathbf{j} \\ \end{array} \end{gathered}$$

In order to understand the movement of two objects after they collide, we need the concept of momentum.

Momentum is a vector property of a moving object. It is a scalar multiple of the velocity of the object, and is given by momentum = mass times velocity.

The standard unit for the magnitude of momentum is newton seconds, denoted Ns. To obtain momentum in Ns, your mass must be in kilograms and your velocity in metres per second.

The important property of momentum is that it is conserved in collisions. That is, when objects collide, the total momentum before collision is equal to the total momentum after collision.