Class 12 Math Case Study Vectors with Answers – NCERT Based

Class 12 Math Case Study Vectors with Answers

In a space research lab, engineers are developing a satellite launch simulation model. To program the movement of the satellite in 3D space, they need to calculate the **position and direction of motion** at various stages. Every motion is **resolved into components** along the three coordinate axes. They study how vectors can be expressed in terms of **direction cosines**, direction ratios, and unit vectors. By using the **position vector**, they determine the precise location of the satellite at any instant. Direction cosines help in analyzing the orientation of movement in space. All the engineers must be proficient in converting vectors from geometric to component form in both 2D and 3D, which is essential for accurate navigation.

Theory and Formulae Related to Vector Components:

• Resolution in 3D: $\vec{a} = a_1\hat{i} + a_2\hat{j} + a_3\hat{k}$
• Magnitude: $|\vec{a}| = \sqrt{a_1^2 + a_2^2 + a_3^2}$
• Direction cosines: $\cos\alpha = \frac{a_1}{|\vec{a}|}, \cos\beta = \frac{a_2}{|\vec{a}|}, \cos\gamma = \frac{a_3}{|\vec{a}|}$
• Property: $\cos^2\alpha + \cos^2\beta + \cos^2\gamma = 1$
• Position vector of a point $P(x, y, z)$ is $\vec{OP} = x\hat{i} + y\hat{j} + z\hat{k}$