Question

A particle undergoes a displacement Δr⃗ of magnitude 54 m in a direction 42 ∘ below the x axis. Express Δr⃗ in terms of the unit vectors x^ and y^.

Answer 1

There are several ways to write this vector:
 Form 1: 
 Angle measured clockwise from the x axis
 Δr = 54 * (cos (42) x ^ - sin (42) y ^)
 Form 2: 
 Angle measured counterclockwise from the x axis
 Δr = 54 * (cos (318) x ^ + sin (318) y ^)
 answer
 Δr = 54 * (cos (42) x ^ - sin (42) y ^)
 Δr = 54 * (cos (318) x ^ + sin (318) y ^)

Answer 2

The displacement of the particle in terms of its unit vectors is given by $\boxed{\Delta r=(40.13\text{ m})\,\hat x-(36.13\text{ m})\hat y}$.

Explanation:

Given:

The magnitude of displacement of the particle is $54\text{ m}$.

The angle made by the particle with the X-axis is $42^\circ$ in negative direction.

Concept:

The displacement of a body is defined as the shortest distance between the initial and the final position of the body. The displacement of a body includes the direction along with the magnitude of displacement of the body. Therefore, displacement is termed as the vector quantity.

The displacement of the particle is in the positive X-direction but making an angle of $42^\circ$ below the X-axis. Therefore, the overall displacement of the particle is in the positive direction of X-axis and negative direction of Y-axis.

The displacement of the particle is represented as:

$\boxed{\Delta r=A\,cos\,42^\circ\,\hat x\,-A\,sin\,42^\circ\,\hat y}$

Here, $A$ is the magnitude of displacement of the particle.

Substitute the value of displacement in above expression.

$\begin{aligned}\Delta r&=54\,cos\,42^\circ\,\hat x\,-54\,sin\,42^\circ\,\hat y\\&=40.13\text{ m}\,\hat x-36.13\text{ m}\,\hat y\end{aligned}$

Thus, The displacement of the particle in terms of its unit vectors is given by $\boxed{\Delta r=(40.13\text{ m})\,\hat x-(36.13\text{ m})\hat y}$.

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Answer Details:

Grade: Senior School

Subject: Physics

Chapter: Vectors and Scalars

Keywords:

particle, displacement, magnitude, 54 m, x-axis, direction, negative, y-axis, unit vectors, vector, shortest distance.