Explore advanced tenets of vectors (ratio theorems, dot product, cross product, vector differentiation, and vector integration)

The dot product of two vectors $\underline{a} = (a_1, a_2, a_3)$ and $\underline{b} = (b_1, b_2, b_3)$ is defined as: $\underline{a} \cdot \underline{b} = a_1b_1 + a_2b_2 + a_3b_3.$

It is also related to the angle $\theta$ between them by: $\underline{a} \cdot \underline{b} = |\underline{a}||\underline{b}|\cos\theta \quad \text{and} \quad \theta = \cos^{-1}\left(\frac{\underline{a} \cdot \underline{b}}{|\underline{a}||\underline{b}|}\right).$

Properties:

• $\underline{a}$ and $\underline{b}$ are perpendicular (orthogonal) if and only if $\underline{a} \cdot \underline{b} = 0$.
• Work Done: If a force $\underline{F}$ displaces an object through $\underline{d}$, the work done is $W = \underline{F} \cdot \underline{d}$.

Example: Find the work done by a force $\underline{F} = 2\underline{i} + \underline{j} + 3\underline{k}$ moving a particle from point $A(2,2,3)$ to $B(3,4,5)$. Displacement $\underline{d} = \overrightarrow{AB} = \underline{i} + 2\underline{j} + 2\underline{k}$. $W = (2)(1) + (1)(2) + (3)(2) = 10$ units.

The cross product of $\underline{a}$ and $\underline{b}$ results in a vector perpendicular to both, with magnitude $|\underline{a}||\underline{b}|\sin\theta$. It is calculated using the determinant: $\underline{a} \times \underline{b} = \begin{vmatrix} \underline{i} & \underline{j} & \underline{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \end{vmatrix}.$

Applications:

• Area of a Parallelogram: $A = |\underline{a} \times \underline{b}|$.
• Area of a Triangle: $A = \frac{1}{2}|\underline{a} \times \underline{b}|$.

Example: Find the area of a triangle with vertices $P_1(2,3,4)$, $P_2(1,1,1)$, $P_3(3,0,-1)$. Vectors: $\overline{P_1P_2} = -\underline{i} - 2\underline{j} - 3\underline{k}$, $\overline{P_1P_3} = \underline{i} - 3\underline{j} - 5\underline{k}$. $\overline{P_1P_2} \times \overline{P_1P_3} = \underline{i} - 8\underline{j} + 5\underline{k}$. Area $= \frac{1}{2}\sqrt{1^2 + (-8)^2 + 5^2} = \frac{1}{2}\sqrt{90} = \frac{3\sqrt{10}}{2}$ square units.

If a particle's position is given by $\underline{r}(t) = f(t)\underline{i} + g(t)\underline{j} + h(t)\underline{k}$, then:

• Velocity: $\underline{v}(t) = \frac{d\underline{r}}{dt} = f'(t)\underline{i} + g'(t)\underline{j} + h'(t)\underline{k}$.
• Acceleration: $\underline{a}(t) = \frac{d\underline{v}}{dt} = \frac{d^2\underline{r}}{dt^2}$.

Example: If $\underline{r}(t) = (2t+3)\underline{i} + (1-t)\underline{j} + (1-5t^2)\underline{k}$, find the velocity. $\underline{v} = \frac{d\underline{r}}{dt} = 2\underline{i} - \underline{j} - 10t\underline{k}$.

Integration is the reverse process of differentiation. If $\underline{v}(t)$ is velocity, the position is found by integrating: $\underline{r}(t) = \int \underline{v}(t) \, dt = \left[\int f(t)dt\right]\underline{i} + \left[\int g(t)dt\right]\underline{j} + \left[\int h(t)dt\right]\underline{k} + \underline{C}.$ (Where $\underline{C}$ is a constant vector determined by initial conditions).

Example: Given acceleration $\underline{a} = 2\sin 2t\underline{i} - 3\cos t\underline{j} + t\underline{k}$ and initial velocity $\underline{v}(0) = 4\underline{i} + 2\underline{j}$, find $\underline{v}(t)$. Integrate $\underline{a}$: $\underline{v}(t) = \int (2\sin 2t)dt\underline{i} + \int (-3\cos t)dt\underline{j} + \int t dt\underline{k} + \underline{C}$. $\underline{v}(t) = -\cos 2t\underline{i} - 3\sin t\underline{j} + \frac{1}{2}t^2\underline{k} + \underline{C}_1$. Using $\underline{v}(0) = -\underline{i} + \underline{C}_1 = 4\underline{i} + 2\underline{j}$, we find $\underline{C}_1 = 5\underline{i} + 2\underline{j}$. Thus, $\underline{v}(t) = (5 - \cos 2t)\underline{i} + (2 - 3\sin t)\underline{j} + \frac{1}{2}t^2\underline{k}$.

In Tanzanian engineering and construction projects, such as building the Kigamboni Bridge in Dar es Salaam, vectors are essential for calculating structural forces. Engineers use dot products to determine how forces align with structural beams (work done) and cross products to calculate the area of cross-sections and the torque exerted by winds or water currents, ensuring the stability of the structure.