Given that a particle undergoes a displacement of magnitude $54 m$ in a direction $42°$ below the x-axis, what are the x and y components of the displacement vector?

• A. x-component: $-54\times cos(42°)$, y-component: $-54\times sin(42°)$
• B. x-component: $54\times cos(42°)$, y-component: $-54\times sin(42°)$
• C. x-component: $54\times cos(42°)$, y-component: $+54\times sin(42°)$
• D. x-component: $54\times sin(42°)$, y-component: $+54\times cos(42°)$

Answer: (B)

Verified step by step guidance

1. Identify the magnitude of the displacement vector, which is given as 54 m, and the direction, which is 42° below the x-axis. This means the angle is measured downward from the positive x-axis.
2. Recall that the x-component of a vector is found by multiplying the magnitude by the cosine of the angle: \(x = 54 \times \cos(42^\circ)\).
3. Recall that the y-component of a vector is found by multiplying the magnitude by the sine of the angle. Since the direction is below the x-axis, the y-component will be negative: \(y = -54 \times \sin(42^\circ)\).
4. Write the displacement vector components as \(\vec{d} = (54 \times \cos(42^\circ), -54 \times \sin(42^\circ))\) to represent the correct direction and magnitude.
5. Verify the signs of the components based on the quadrant: since the vector is below the x-axis and to the right, the x-component is positive and the y-component is negative.