RD Sharma Class 12 Solutions Chapter 27 - Direction Cosines and Direction Ratios (Ex 27.1) Exercise 27.1 - Free PDF
FAQs on RD Sharma Class 12 Solutions Chapter 27 - Direction Cosines and Direction Ratios (Ex 27.1) Exercise 27.1
1. What is the step-by-step method to find the direction cosines of a line if its direction ratios (a, b, c) are given, as often required in RD Sharma Ex 27.1?
To find the direction cosines (l, m, n) from direction ratios (a, b, c), follow these steps as per the correct methodology:
First, calculate the magnitude, r = √(a² + b² + c²).
Then, divide each direction ratio by this magnitude to find the corresponding direction cosine.
The formulas are: l = a/r, m = b/r, and n = c/r.
Remember that a line can have two sets of direction cosines: (l, m, n) and (-l, -m, -n), representing its two opposite directions.
2. How do you correctly solve problems that require finding the direction cosines of a line joining two points, P(x₁, y₁, z₁) and Q(x₂, y₂, z₂)?
The correct procedure involves two main stages:
Find the Direction Ratios: First, determine the direction ratios (a, b, c) of the line segment PQ by subtracting the coordinates of the initial point from the final point. So, a = (x₂ – x₁), b = (y₂ – y₁), and c = (z₂ – z₁).
Calculate the Direction Cosines: Use the direction ratios found in the previous step and apply the standard formula. Calculate the distance PQ = √((x₂ – x₁)² + (y₂ – y₁)² + (z₂ – z₁)²), and then the direction cosines are l = (x₂ – x₁)/PQ, m = (y₂ – y₁)/PQ, and n = (z₂ – z₁)/PQ.
3. What is the fundamental difference between direction cosines and direction ratios, and why are both concepts important for this chapter?
The core difference lies in their uniqueness and properties:
Direction Cosines (DCs): These are the cosines of the angles that a directed line makes with the positive x, y, and z axes. For a given directed line, the direction cosines (l, m, n) are unique and satisfy the relation l² + m² + n² = 1. They essentially define the orientation of a unit vector along the line.
Direction Ratios (DRs): These are any three numbers (a, b, c) that are proportional to the direction cosines. This means a = kl, b = km, c = kn for some non-zero constant k. An infinite number of direction ratios can exist for a single line. They are often easier to calculate from two points on the line.
Both are important because DRs are simple to find, while DCs are needed for precise geometric calculations like finding the angle between two lines.
4. How can you use direction ratios to prove that three given points A, B, and C are collinear, a common problem in Exercise 27.1?
To prove collinearity using direction ratios, you must show that the line segments AB and BC are parallel and share a common point (B). The steps are:
Calculate the direction ratios of the line segment AB. Let these be (a₁, b₁, c₁).
Calculate the direction ratios of the line segment BC. Let these be (a₂, b₂, c₂).
Check if the direction ratios are proportional. If a₁/a₂ = b₁/b₂ = c₁/c₂, then the line segments AB and BC are parallel. Since they share the common point B, the points A, B, and C must lie on the same straight line, meaning they are collinear.
5. What is the geometric significance of the property l² + m² + n² = 1 for direction cosines?
The property l² + m² + n² = 1 has a clear geometric meaning. The direction cosines (l, m, n) are defined as cos α, cos β, and cos γ respectively. These are the components of a unit vector that lies along the given directed line. The magnitude of any unit vector is, by definition, 1. The formula for the magnitude of a vector with components (l, m, n) is √(l² + m² + n²). Therefore, setting this magnitude to 1 gives √(l² + m² + n²) = 1, which simplifies to l² + m² + n² = 1. It fundamentally states that the direction cosines are coordinates of a point on a unit sphere.
6. If a line is equally inclined to the coordinate axes, what is the correct method to find its direction cosines?
If a line is equally inclined to the axes, it means the angles it makes with the x, y, and z axes are equal (α = β = γ). This implies their cosines are also equal, so l = m = n. To find their value, use the fundamental identity:
Start with l² + m² + n² = 1.
Substitute l for m and n: l² + l² + l² = 1, which simplifies to 3l² = 1.
Solve for l: l² = 1/3, so l = ±1/√3.
Therefore, the direction cosines of the line are (1/√3, 1/√3, 1/√3) or (-1/√3, -1/√3, -1/√3).
7. What is a common mistake students make when finding direction cosines from a given vector?
A frequent mistake is confusing the coefficients of the vector with the direction cosines directly. If a vector is given as v = aî + bĵ + c k̂, the numbers (a, b, c) are its direction ratios, not its direction cosines. To find the correct direction cosines, you must first find the magnitude of the vector, |v| = √(a² + b² + c²), and then divide each component by this magnitude. Forgetting to normalize the vector by dividing by its magnitude is the most common error.
8. How do the concepts in this chapter help determine if two lines are perpendicular?
The concepts of direction cosines and direction ratios provide a straightforward condition for perpendicularity. Two lines are perpendicular if the angle between them is 90°, which means cos 90° = 0.
Using direction cosines (l₁, m₁, n₁) and (l₂, m₂, n₂), the condition for perpendicularity is l₁l₂ + m₁m₂ + n₁n₂ = 0.
Using direction ratios (a₁, b₁, c₁) and (a₂, b₂, c₂), the condition is a₁a₂ + b₁b₂ + c₁c₂ = 0.
This test is a direct application of the dot product of their direction vectors being zero and is a crucial tool for solving problems in three-dimensional geometry.
