Math Case Study Questions on Vectors Class 12 Solved Examples

Math Case Study Questions on Vectors Class 12

In a robotics competition, a team builds a robotic arm that moves in three-dimensional space. To program **rotations and torque**, the students need to understand the **vector cross product** (or vector product). They study how the cross product gives a vector **perpendicular** to two given vectors and how its magnitude can represent the **area of a parallelogram** or triangle. Using the **right-hand rule** , they determine the direction of rotation. They also analyze the **torque** $\vec{\tau} = \vec{r} \times \vec{F}$ and how vector product plays a role in rotational physics. Understanding cross product allows them to model forces and movements accurately in 3D space.

Theory and Formulae Related to Cross Product:

• **Geometric Form**: $\vec{a} \times \vec{b} = |\vec{a}||\vec{b}|\sin\theta\,\hat{n}$ ($\hat{n}$ is the unit vector perpendicular to $\vec{a}$ and $\vec{b}$)
• **Component (Determinant) Form**: $$\vec{a} \times \vec{b} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ a_1 & a_2 & a_3 \\ b_1 & b_2 & b_3 \\ \end{vmatrix} = (a_2b_3 – a_3b_2)\hat{i} – (a_1b_3 – a_3b_1)\hat{j} + (a_1b_2 – a_2b_1)\hat{k}$$
• **Area of parallelogram**: $|\vec{a} \times \vec{b}|$
• **Area of triangle**: $\frac{1}{2}|\vec{a} \times \vec{b}|$
• **Parallelism/Collinearity**: If $\vec{a} \times \vec{b} = \vec{0}$, then vectors are parallel or collinear.