Physics 206 Homework Assignment 10 Due November 8

Phy 206 Due Friday November 8 2013homework Assignment 10problem 1a

Extracted and clarified assignment instructions: Show that the length of the vector r̂ = cos(θ) x̂ + sin(θ) ŷ is 1. Show that the length of θ̂ = -sin(θ) x̂ + cos(θ) ŷ is 1. Show that r̂ and θ̂ are perpendicular. Show that r̂ × θ̂ = ẑ. Sketch r̂ and θ̂ at the points labeled A and B in the diagram.

Paper For Above instruction

In classical vector analysis, the unit vectors associated with Cartesian coordinates are fundamental in understanding directional properties and transformations. Specifically, when analyzing two-dimensional vectors, defining orthogonal unit vectors such as r̂ and θ̂ enables us to explore relations pertinent to polar and cylindrical coordinate systems, which are crucial in fields like physics and engineering.

Part A of the problem asks us to demonstrate that the vector r̂ = cos(θ) x̂ + sin(θ) ŷ is a unit vector with a magnitude of 1. To achieve this, we compute the magnitude |r̂| by taking the square root of the sum of squares of its components:

|r̂| = √[(cos(θ))^2 + (sin(θ))^2] = √[cos^2(θ) + sin^2(θ)].

Using the Pythagorean identity, cos^2(θ) + sin^2(θ) = 1, we find that |r̂| = √1 = 1, confirming that r̂ is a unit vector.

In Part B, the vector θ̂ = -sin(θ) x̂ + cos(θ) ŷ is also a unit vector. Its magnitude is similarly calculated:

|θ̂| = √[(-sin(θ))^2 + (cos(θ))^2] = √[sin^2(θ) + cos^2(θ)] = √1 = 1.

Thus, θ̂ is orthogonal to r̂ and also of unit length.

Part C examines whether r̂ and θ̂ are perpendicular. The dot product r̂ · θ̂ provides insight:

r̂ · θ̂ = (cos(θ))(−sin(θ)) + (sin(θ))(cos(θ)) = −cos(θ) sin(θ) + sin(θ) cos(θ) = 0.

Since their dot product is zero, r̂ and θ̂ are orthogonal, fulfilling the condition for perpendicular vectors.

In Part D, the cross product r̂ × θ̂ is evaluated to confirm whether it equals ẑ:

r̂ × θ̂ = (cos(θ) x̂ + sin(θ) ŷ) × (−sin(θ) x̂ + cos(θ) ŷ).

Expanding this, and utilizing the properties of cross products:

r̂ × θ̂ = cos(θ)(−sin(θ))(x̂ × x̂) + cos(θ)cos(θ)(x̂ × ŷ) + sin(θ)(−sin(θ))(ŷ × x̂) + sin(θ)cos(θ)(ŷ × ŷ).

Since x̂ × x̂ = 0, ŷ × ŷ = 0, and x̂ × ŷ = ẑ, ŷ × x̂ = -ẑ, the expression simplifies to:

r̂ × θ̂ = 0 + cos^2(θ) ẑ + sin^2(θ) ẑ + 0 = (cos^2(θ) + sin^2(θ)) ẑ = 1 ẑ = ẑ.

Therefore, the cross product of r̂ and θ̂ equals ẑ, confirming that they form a right-handed coordinate system.

Lastly, the sketching of r̂ and θ̂ at points A and B would involve drawing vectors originating at the specified points with directions corresponding to the angles θ, illustrating their orthogonal relations and unit length.

References

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