Vectors Notes

Notes for Vectors

Basic vector algebra:

Commutivity of vector addition:

The geometry of a parallelogram can be used to prove that vector addition is commutative:

$$\boxed{\therefore\mathbf{c} = \mathbf{a} + \mathbf{b} = \mathbf{b} + \mathbf{a}}$$

Non-commutivity of vector subtraction:

The geometry of a triangle can be used to prove that vector subtraction is non-commutative:

$$\boxed{\mathbf{\therefore c} = \mathbf{b} - \mathbf{a} = - (\mathbf{a} - \mathbf{b})}$$

Associativity of vector addition:

The geometry of a parallelepiped can be used to prove that vector addition is associative:

The ratio theorem:

The ratio theorem is a method for solving geometric problems using vectors, by expressing the vectors as a ratio of a third vector:

Therefore, $\overrightarrow{OX}$ can be expressed in terms of $\overrightarrow{OA}$, $\overrightarrow{OB}$, λ and μ:

$$\frac{\overrightarrow{AX}}{\overrightarrow{XB}} = \frac{\lambda}{\mu}\therefore\overrightarrow{OX} - \overrightarrow{OA} = \frac{\lambda}{\mu}\left( \overrightarrow{OX} + \overrightarrow{OB} \right)$$

$$\therefore\mu\left( \overrightarrow{OX} - \overrightarrow{OA} \right) = \lambda\left( \overrightarrow{OX} + \overrightarrow{OB} \right)\therefore(\mu - \lambda)\overrightarrow{OX} = \mu\overrightarrow{OA} + \lambda\overrightarrow{OB}$$

$$\mathbf{\therefore}\overrightarrow{\mathbf{OX}}\mathbf{=}\frac{\mathbf{\mu}\overrightarrow{\mathbf{OA}}\mathbf{+ \lambda}\overrightarrow{\mathbf{OB}}}{\left( \mathbf{\mu - \lambda} \right)}$$

Where, if the total length of $\overrightarrow{AB}$ is taken to be 1, then μ=1λ such that $\overrightarrow{OX}$ can be expressed in terms of $\overrightarrow{OA}$, $\overrightarrow{OB}$ and λ only.

Using the ratio theorem to prove that the diagonals of a parallelogram bisect:

$$\overrightarrow{OA} = \overrightarrow{OX} + \overrightarrow{XA} = \lambda\overrightarrow{OC} - \mu\overrightarrow{AB}$$

$$\overrightarrow{BC} = \overrightarrow{BX} + \overrightarrow{XC} = (1 - \lambda)\overrightarrow{OC} - (1 - \mu)\overrightarrow{AB}$$

$$\overrightarrow{OA} = \overrightarrow{BC}$$

$$\therefore\lambda\overrightarrow{OC} - \mu\overrightarrow{AB} = (1 - \lambda)\overrightarrow{OC} - (1 - \mu)\overrightarrow{AB}$$

Comparing coefficients:

λ=1λ,μ=1μ

$$\therefore 2\lambda = 1,\ \ 2\mu = 1\mathbf{\therefore\lambda = \mu =}\frac{\mathbf{1}}{\mathbf{2}}\blacksquare$$

Vectors in kinematics:

The kinematic quantities of displacement, velocity and acceleration have both magnitude and direction, hence are vector quantities. They are related by vector calculus as follows:

Displacement: For a moving object, displacement is a function of time, r(t).

Velocity: Velocity is the rate of change of displacement:

$$\mathbf{v}(t) = \lim_{\delta t \rightarrow 0}\left( \frac{\mathbf{r}(t + \delta t) - \mathbf{r}(t)}{\delta t} \right) = \lim_{\delta t \rightarrow 0}\frac{\delta\mathbf{r}}{\delta t} = \frac{d\mathbf{r}}{dt} = \dot{\mathbf{r}}(t)$$

$$\boxed{\therefore\mathbf{v}(t) = \dot{\mathbf{r}}(t)}$$

Acceleration: Acceleration is the rate of change of velocity:

$$\mathbf{a}(t) = \lim_{\delta t \rightarrow 0}\left( \frac{\mathbf{v}(t + \delta t) - \mathbf{v}(t)}{\delta t} \right) = \lim_{\delta t \rightarrow 0}\frac{\delta\mathbf{v}}{\delta t} = \frac{d\mathbf{v}}{dt} = \dot{\mathbf{v}}(t) = \ddot{\mathbf{r}}(t)$$

$$\boxed{\therefore\mathbf{a}(t) = \dot{\mathbf{v}}(t) = \ddot{\mathbf{r}}(t)}$$

Since the differentiation of vector functions may change their direction, the displacement, velocity and acceleration of a moving object are not necessarily parallel.

Scalar quantities in kinematics:

The kinematic quantities of distance and speed are the cumulative scalar analogues to displacement and velocity. The scalar equivalent to acceleration can be found using the scalar product, and will therefore be explored later:

Distance: Since distance is a cumulative quantity, it cannot be directly related to the displacement function r(t); only to an incremental change in r(t), denoted δr. Since δr is very small, it can be approximated as a straight line, hence the cumulative nature of distance is not significant:

$$\boxed{distance = \left| \delta\mathbf{r} \right|}$$

Speed: Speed is the rate of change of distance:

$$speed = \lim_{\delta t \rightarrow 0}\left( \frac{\left| \delta\mathbf{r} \right|}{\delta t} \right) = \lim_{\delta t \rightarrow 0}\left( \frac{\left| \delta t\mathbf{v} \right|}{\delta t} \right) = \lim_{\delta t \rightarrow 0}\left( \left| \mathbf{v} \right| \right) = \left| \mathbf{v} \right|$$

$$\boxed{\therefore speed = \left| \mathbf{v} \right|}$$

Geometric vector equations:

Through the use of vectors, it is possible to generate equations to represent any line, plane or sphere, at any point in three-dimensional space.

Vector equation of a line:

A line is one-dimensional and is therefore defined along only one direction in three-dimensional space. Thus, the vector equation of a line contains only one direction vector, along with a point vector to specify its position:

$$\boxed{\mathbf{r = a +}\lambda\mathbf{(b - a)}}$$

The position vector a defines the position of the line at a point A in space, while the second vector λ(ba) defines an arbitrary distance λ along the vector joining points A and B, therefore specifies the direction of the line.

r is therefore the position vector of any point on the line from the origin.

Vector equation of a plane:

A plane is two-dimensional and is therefore defined along two directions in three-dimensional space. Thus, the vector equation of a plane contains two direction vectors, along with a point vector to specify its position:

$$\boxed{\mathbf{r = a +}\lambda\left( \mathbf{b - a} \right)\mathbf{+}\mu\mathbf{(c - a)}}$$

The position vector a defines the position of the plane at a point A in space; the second vector λ(ba) defines an arbitrary distance λ along the vector joining points A and B, and the third vector μ(ca) defines an arbitrary distance μ along the vector joining points A and C. The vectors (ba) and...

Buy the full version of these notes or essay plans and more in our Mathematics for Natural Sciences Notes.