if vectors a=coswt i + sinwt j ,Re,if vectors a=coswt i + sinwt j, If vectors A = cos wt i + Sin wt j and B = cos wt/2 i + sin wt/2 j are functions of time , then the value of t at which they are orthogonal to. LOUIS VUITTON Official International site - Discover our latest Jewelry's Blossom Collections collection, exclusively on louisvuitton.com and in Louis Vuitton Stores
In the realm of vector calculus and physics, time-dependent vectors play a crucial role in describing motion and dynamic systems. This article delves into the specific case of vectors of the form a = cos(ωt) i + sin(ωt) j, where ω represents angular frequency and t denotes time. We will explore the properties of such vectors and, more specifically, investigate the conditions under which two vectors of this form become orthogonal. We will particularly focus on the scenario where we have two vectors, A = cos(ωt) i + sin(ωt) j and B = cos(ωt/2) i + sin(ωt/2) j, and determine the time 't' at which they are perpendicular to each other. This exploration will involve understanding the concept of orthogonality, the dot product of vectors, and solving trigonometric equations.
Understanding the Vectors
Before diving into the orthogonality condition, let's first understand the nature of the vector a = cos(ωt) i + sin(ωt) j. This vector is a two-dimensional vector that lies in the xy-plane. The components of the vector are cos(ωt) and sin(ωt), which are functions of time. As time 't' progresses, the vector changes its direction while maintaining a constant magnitude.
* Magnitude: The magnitude of the vector a is given by:
|a| = √(cos²(ωt) + sin²(ωt)) = √1 = 1
This reveals that the magnitude of the vector is always 1, regardless of the value of 't'. This signifies that the vector is a unit vector.
* Direction: The direction of the vector is determined by the angle it makes with the positive x-axis. This angle, let's call it θ, is given by:
θ = arctan(sin(ωt) / cos(ωt)) = arctan(tan(ωt)) = ωtif vectors a=coswt i + sinwt j
Therefore, the angle that the vector makes with the x-axis is directly proportional to time 't' and the angular frequency ω. This means that the vector rotates counter-clockwise with a constant angular velocity ω.
In essence, the vector a = cos(ωt) i + sin(ωt) j represents a point moving along a unit circle in the xy-plane with a constant angular speed ω.
Similarly, the vector B = cos(ωt/2) i + sin(ωt/2) j also represents a unit vector rotating in the xy-plane. However, its angular speed is ω/2, which is half the angular speed of vector A.
Orthogonality and the Dot Product
Two vectors are said to be orthogonal or perpendicular if the angle between them is 90 degrees (π/2 radians). A fundamental property that links orthogonality to vector algebra is the dot product. The dot product of two vectors is defined as the product of their magnitudes and the cosine of the angle between them:
A ⋅ B = |A| |B| cos(θ)
where θ is the angle between vectors A and B.
If A and B are orthogonal, then θ = π/2, and cos(π/2) = 0. Consequently, the dot product of two orthogonal vectors is zero:
A ⋅ B = 0 (for orthogonal vectors)
Conversely, if the dot product of two non-zero vectors is zero, then the vectors are orthogonal. This provides a convenient algebraic method to check for orthogonality.
Applying the Orthogonality Condition to Vectors A and B
Now, let's apply this principle to our vectors A = cos(ωt) i + sin(ωt) j and B = cos(ωt/2) i + sin(ωt/2) j. To find the time 't' at which they are orthogonal, we need to find the value of 't' for which their dot product is zero:
A ⋅ B = (cos(ωt) i + sin(ωt) j) ⋅ (cos(ωt/2) i + sin(ωt/2) j) = 0
The dot product of two vectors in component form is calculated as the sum of the products of their corresponding components:
A ⋅ B = cos(ωt)cos(ωt/2) + sin(ωt)sin(ωt/2) = 0
This equation represents the condition for orthogonality between vectors A and B. We need to solve this trigonometric equation for 't'.
Solving the Trigonometric Equation
The equation cos(ωt)cos(ωt/2) + sin(ωt)sin(ωt/2) = 0 can be simplified using the trigonometric identity:
cos(A - B) = cos(A)cos(B) + sin(A)sin(B)
Applying this identity to our equation, we get:
cos(ωt - ωt/2) = 0
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if vectors a=coswt i + sinwt j - Re