A man pushing a mop across a floor causes it to undergo two...

A man pushing a mop across a floor causes it to undergo two displacements. The first has a magnitude of $150 \mathrm{~cm}$ and makes an angle of $120^{\circ}$ with the positive $x$ axis. The resultant displacement has a magnitude of $140 \mathrm{~cm}$ and is directed at an angle of $35.0^{\circ}$ to the positive $x$ axis. Find the magnitude and direction of the second displacement.

Key Concepts

Vectors

Vectors are quantities that have both magnitude and direction. They are used to represent quantities such as displacement, velocity, and force. In problems like these, each displacement is treated as a vector, and its magnitude and angle determine its representation in a coordinate system.

Vector Components

Decomposing a vector into its horizontal (x-axis) and vertical (y-axis) components allows for easier manipulation during calculations. This is done using trigonometric functions (cosine and sine) based on the vector's magnitude and the angle it makes with the positive x-axis.

Vector Addition

Vector addition involves summing the corresponding components of each vector. In the context of displacement, the resultant displacement is obtained by adding all individual displacement vectors together, which yields a new vector representing the overall change in position.

Resultant Displacement

The resultant displacement is the vector sum of all individual displacements. It is characterized by a specific magnitude and direction. Finding the resultant involves using the calculated components from the individual vectors and recombining them to determine the overall effect.

Vector Subtraction

When one vector is unknown but the resultant and another vector are known, vector subtraction can be used to find the unknown vector. This is performed by subtracting the known vector's components from the resultant vector's components, yielding the magnitude and direction of the second displacement.

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