Math Case Study Vectors with Answers PDF Class 12 Solutions

Math Case Study Vectors with Answers PDF Class 12 Solutions

A group of engineering students are working on a robotics project that involves controlling a robot’s path using vector inputs. They define the robot’s movement in terms of vector displacements. The path consists of sequential displacements represented as **vectors**. They use **vector addition** to compute the resultant path, **scalar multiplication** to change the speed (magnitude) of the movement, and **subtraction** to find the change in direction. The team learns that the **Triangle Law** of vector addition helps combine vectors when applied head-to-tail, while the **Parallelogram Law** is useful when vectors are applied from the same point. They apply vector algebra in two and three dimensions to program the robot’s navigation efficiently.

Theory and Formulae Related to Algebra of Vectors:

• Vector addition: $\vec{a} + \vec{b} = (a_1 + b_1)\hat{i} + (a_2 + b_2)\hat{j} + (a_3 + b_3)\hat{k}$
• Vector subtraction: $\vec{a} – \vec{b} = (a_1 – b_1)\hat{i} + (a_2 – b_2)\hat{j} + (a_3 – b_3)\hat{k}$
• Scalar multiplication: $k\vec{a} = (ka_1)\hat{i} + (ka_2)\hat{j} + (ka_3)\hat{k}$
• **Triangle Law**: $\vec{a} + \vec{b}$ is the third side of triangle formed when vectors are placed head to tail.
• **Parallelogram Law**: Resultant of $\vec{a}$ and $\vec{b}$ is the diagonal of the parallelogram formed by $\vec{a}$ and $\vec{b}$ from same initial point. [Image of Triangle Law and Parallelogram Law of Vector Addition]