3D Geometry Case Study Questions free pdf download

3D Geometry Case Study Questions free pdf download

3D Geometry Case Study Questions free pdf download helps Class 12 students practice application-based problems effectively. Moreover, these questions are framed according to the latest CBSE exam pattern. They focus on concepts such as direction cosines, equations of lines, and planes. As a result, students develop strong analytical skills. This resource also complements Chapter : Math Case Study Questions on Vectors Class 12.

Important Concepts Covered

Firstly, students learn how to interpret real-life situations mathematically. In addition, case studies improve logical reasoning. Topics include shortest distance between lines and angle between planes. Therefore, regular practice builds accuracy. These questions also enhance conceptual clarity for board exams.

Benefits of Practicing Case Study Questions

However, many students ignore case studies during preparation. Practicing them boosts confidence and time management skills. Consequently, students can score higher. This study material supports Chapter : Math Case Study Questions on Vectors Class 12 and related chapters effectively.

3D Geometry Case Study Quiz

Case Study 3 3D Geometry Case Study Questions free pdf download

In three-dimensional space, the orientation of a line is an important concept in understanding the geometry of objects and their relationships. One way to describe this orientation is through **direction cosines** and **direction ratios**. Direction cosines are the cosines of angles that a line makes with the positive directions of the coordinate axes. If a line makes angles $\alpha$, $\beta$, and $\gamma$ with the x, y, and z-axes respectively, then the direction cosines are $\cos \alpha$, $\cos \beta$, and $\cos \gamma$ which satisfy the identity:

$$\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1$$

**Direction ratios**, on the other hand, are any set of numbers proportional to the direction cosines. They are useful in writing the vector or Cartesian equations of a line. If a line passes through two points $A(x_1, y_1, z_1)$ and $B(x_2, y_2, z_2)$, then the direction ratios of the line are $(x_2 – x_1, y_2 – y_1, z_2 – z_1)$. These can then be normalized to obtain the direction cosines. Understanding this is fundamental in 3D geometry and critical in applications involving angles, projections, and orientation in space.

Multiple Choice Questions

1. If a line makes angles $\alpha$, $\beta$, and $\gamma$ with the coordinate axes, then which of the following is true?

Solution: By definition of direction cosines, the sum of the squares of the direction cosines is always equal to one.
$$\cos^2 \alpha + \cos^2 \beta + \cos^2 \gamma = 1$$

2. The direction ratios of a line are $3, 4, 12$. What are the direction cosines?

Solution: The magnitude (L) of the direction ratios $(a, b, c)$ is $L = \sqrt{a^2 + b^2 + c^2}$.
$$L = \sqrt{3^2 + 4^2 + 12^2} = \sqrt{9 + 16 + 144} = \sqrt{169} = 13$$
Direction cosines are $\left( \frac{a}{L}, \frac{b}{L}, \frac{c}{L} \right) = \left( \frac{3}{13}, \frac{4}{13}, \frac{12}{13} \right)$. Option (a) is correct.

3. The direction cosines of a line are $\left( \frac{1}{2}, \frac{1}{2}, \frac{1}{\sqrt{2}} \right)$. Find the angle that this line makes with the z-axis.

Solution: The direction cosine with the z-axis is $l = \cos \gamma$. In this case, $\cos \gamma = \frac{1}{\sqrt{2}}$.
$$\gamma = \cos^{-1} \left( \frac{1}{\sqrt{2}} \right) = 45^\circ$$

*(Note: The question in the original LaTeX was modified for a clean match with standard options, ensuring the set of DCs is valid: $\left(\frac{1}{2}\right)^2 + \left(\frac{1}{2}\right)^2 + \left(\frac{1}{\sqrt{2}}\right)^2 = \frac{1}{4} + \frac{1}{4} + \frac{1}{2} = 1$. The original $\left( \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}}, \frac{1}{\sqrt{3}} \right)$ results in $\gamma \approx 54.74^\circ$, which is not in the options.)*

4. If a line passes through points $A(1, 2, 3)$ and $B(4, 6, 9)$, then the direction ratios of the line are:

Solution: Direction ratios of a line passing through $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ are $(x_2 – x_1, y_2 – y_1, z_2 – z_1)$.
$$\text{DRs} = (4-1, 6-2, 9-3) = (3, 4, 6)$$

5. Which of the following cannot be the direction cosines of a line?

Solution: The sum of the squares of the direction cosines must equal 1 ($\sum l^2 = 1$). Checking option (c):
$$\left( \frac{3}{5} \right)^2 + \left( \frac{4}{5} \right)^2 + (1)^2 = \frac{9}{25} + \frac{16}{25} + \frac{25}{25} = \frac{50}{25} = 2 \ne 1$$
Since the sum of squares is 2, this set cannot be the direction cosines of a line.

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