Class 12 Case Study Questions on Vectors with Answers for Board Exam

Class 12 Case Study Questions on Vectors with Answers

In a mechanics lab, a team is analyzing the motion of a block being pushed along an inclined plane. The **force applied** and the **displacement** are represented by vectors. To calculate the **work done** ($W = \vec{F} \cdot \vec{d}$), the students use the concept of **dot product** (or scalar product). They also explore how the dot product helps find the **angle between two vectors** and the **projection of one vector onto another**. The dot product is useful not only in physics but also in computer graphics, engineering, and navigation systems. By applying formulas, the students calculate work done, identify **orthogonal vectors**, and understand the geometric interpretations of the scalar product.

Theory and Formulae Related to Dot Product:

• **Component Form**: $\vec{a} \cdot \vec{b} = a_1b_1 + a_2b_2 + a_3b_3$
• **Geometric Form**: $\vec{a} \cdot \vec{b} = |\vec{a}||\vec{b}|\cos\theta$ [Image of geometric representation of dot product showing vectors a and b and angle theta]
• **Angle between vectors**: $\cos\theta = \frac{\vec{a} \cdot \vec{b}}{|\vec{a}||\vec{b}|}$
• **Perpendicularity (Orthogonal Vectors)**: If $\vec{a} \cdot \vec{b} = 0$, then $\vec{a} \perp \vec{b}$.
• **Projection of $\vec{a}$ on $\vec{b}$**: $\text{Proj}_{\vec{b}}\vec{a} = \frac{\vec{a} \cdot \vec{b}}{|\vec{b}|}$