Let's break down your questions one by one, starting with the first one regarding the relationship between vectors A, B, C, and D. The equation you provided, A X B = C + D, involves vector operations and their geometric interpretations. To understand why option D is correct, we need to analyze the components of the vectors involved.
Understanding the Vector Equation
The equation A X B = C + D indicates that the cross product of vectors A and B results in the vector sum of C and D. In vector algebra, the cross product of two vectors yields a vector that is perpendicular to both of the original vectors. This means that:
- If B is parallel to C + D, then it suggests that the resultant vector from the cross product A X B is aligned in a direction that is orthogonal to both A and B.
- When we say A is perpendicular to C, it implies that the direction of A does not influence the direction of C, reinforcing the idea of orthogonality.
Now, let's analyze the components:
- The component of C along A can be represented as the projection of C onto A.
- Similarly, the component of D along A is the projection of D onto A.
Given that the component of C along A is equal to the negative of the component of D along A, we can infer that the two components are in opposite directions. This leads us to conclude that the correct option is indeed D, as it aligns with the geometric interpretation of the vectors involved.
Examining the Plane and Vector A
Moving on to your second question about the plane inclined at an angle of 30 degrees with the horizontal and the vector A = -10k. To find the component of vector A that is perpendicular to the inclined plane, we can use trigonometric principles.
Calculating the Perpendicular Component
When a plane is inclined at an angle θ, the component of a vector perpendicular to the plane can be calculated using the formula:
Perpendicular Component = |A| * cos(θ)
In this case:
- Magnitude of A = 10 (since it's -10k, we consider the magnitude)
- Angle θ = 30 degrees
Now, substituting the values:
Perpendicular Component = 10 * cos(30°) = 10 * (√3/2) = 5√3
However, since the question states the answer is 5, it seems there might be a misunderstanding in the interpretation of the components. If we are looking for the component along the direction of the plane, we would use sin(30°) instead, which gives us a different perspective.
Exploring Non-Collinear Unit Vectors
Now, let's tackle your third question regarding the non-collinear unit vectors a1 and a2. You mentioned that |a1 + a2| = √3, and we need to find the value of (a1 - a2)·(2a1 + a2).
Using Vector Properties
We know that:
- |a1| = |a2| = 1 (since they are unit vectors)
- Using the formula for the magnitude of the sum of two vectors, we have:
|a1 + a2|² = |a1|² + |a2|² + 2(a1·a2)
Substituting the known values:
√3² = 1 + 1 + 2(a1·a2)
3 = 2 + 2(a1·a2)
Solving for a1·a2 gives us:
a1·a2 = 1/2
Now, we can compute (a1 - a2)·(2a1 + a2):
(a1 - a2)·(2a1 + a2) = a1·(2a1) + a1·a2 - a2·(2a1) - a2·a2
Substituting the known values:
- a1·a1 = 1
- a2·a2 = 1
- a1·a2 = 1/2
Thus, we have:
2(a1·a1) + a1·a2 - 2(a1·a2) - 1 = 2 - 1/2 - 1 = 1/2
This confirms that the answer is indeed 1/2.
Final Thoughts
Each of these questions involves understanding vector relationships and applying geometric interpretations. By breaking down the components and using trigonometric principles, we can derive the correct answers effectively. If you have any further questions or need clarification on any of these concepts, feel free to ask!
