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If under root 3 magnitude a vector into b vector is equal to a vector point b vector then find the angle between a vector and b vector?
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If under root 3 magnitude a vector into b vector is equal to a vector ...
Understanding the Problem
We have two vectors, a and b, and we know the following relationship:
- The magnitude of the cross product of vectors a and b is equal to the dot product of vectors a and b.
This can be represented mathematically as:
|a x b| = a · b
Exploring Vector Operations
- Magnitude of Cross Product:
The magnitude of the cross product |a x b| is given by |a| |b| sin(θ), where θ is the angle between vectors a and b.
- Dot Product:
The dot product a · b is given by |a| |b| cos(θ).
Setting Up the Equation
From the given condition:
|a| |b| sin(θ) = |a| |b| cos(θ)
Assuming |a| and |b| are not zero, we can simplify this to:
sin(θ) = cos(θ)
Finding the Angle
The equation sin(θ) = cos(θ) implies:
tan(θ) = 1
- Thus, θ = 45 degrees (or π/4 radians)
Conclusion
The angle between vectors a and b is:
- θ = 45 degrees
This angle indicates that the vectors are oriented equally between the axes, which is a unique characteristic of vectors forming an isosceles right triangle in the Cartesian plane.
We have two vectors, a and b, and we know the following relationship:
- The magnitude of the cross product of vectors a and b is equal to the dot product of vectors a and b.
This can be represented mathematically as:
|a x b| = a · b
Exploring Vector Operations
- Magnitude of Cross Product:
The magnitude of the cross product |a x b| is given by |a| |b| sin(θ), where θ is the angle between vectors a and b.
- Dot Product:
The dot product a · b is given by |a| |b| cos(θ).
Setting Up the Equation
From the given condition:
|a| |b| sin(θ) = |a| |b| cos(θ)
Assuming |a| and |b| are not zero, we can simplify this to:
sin(θ) = cos(θ)
Finding the Angle
The equation sin(θ) = cos(θ) implies:
tan(θ) = 1
- Thus, θ = 45 degrees (or π/4 radians)
Conclusion
The angle between vectors a and b is:
- θ = 45 degrees
This angle indicates that the vectors are oriented equally between the axes, which is a unique characteristic of vectors forming an isosceles right triangle in the Cartesian plane.
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If under root 3 magnitude a vector into b vector is equal to a vector point b vector then find the angle between a vector and b vector?
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If under root 3 magnitude a vector into b vector is equal to a vector point b vector then find the angle between a vector and b vector? for UPSC 2025 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about If under root 3 magnitude a vector into b vector is equal to a vector point b vector then find the angle between a vector and b vector? covers all topics & solutions for UPSC 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If under root 3 magnitude a vector into b vector is equal to a vector point b vector then find the angle between a vector and b vector?.
If under root 3 magnitude a vector into b vector is equal to a vector point b vector then find the angle between a vector and b vector? for UPSC 2025 is part of UPSC preparation. The Question and answers have been prepared according to the UPSC exam syllabus. Information about If under root 3 magnitude a vector into b vector is equal to a vector point b vector then find the angle between a vector and b vector? covers all topics & solutions for UPSC 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for If under root 3 magnitude a vector into b vector is equal to a vector point b vector then find the angle between a vector and b vector?.
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