Question

A fly enters through an open window and zooms around the room. In a Cartesian coordinate system with three axes along three edges of the room, the fly changes its position from point b(4.0 m, 1.5 m, 2.5 m) to point e(1.0 m, 4.5 m, 0.5 m). Find the scalar components of the fly’s displacement vector and express its displacement vector in vector component form. What is its magnitude?

a) Scalar components: (-3.0 m, 3.0 m, -2.0 m), Vector component form: (-3.0i + 3.0j - 2.0k) m, Magnitude: 4.358 m
b) Scalar components: (-2.0 m, 3.0 m, 2.0 m), Vector component form: (-2.0i + 3.0j + 2.0k) m, Magnitude: 4.123 m
c) Scalar components: (-5.0 m, 3.0 m, -2.0 m), Vector component form: (-5.0i + 3.0j - 2.0k) m, Magnitude: 6.708 m
d) Scalar components: (-3.0 m, 5.0 m, -2.0 m), Vector component form: (-3.0i + 5.0j - 2.0k) m, Magnitude: 6.708 m

Answer 1

The scalar components of the fly's displacement vector are (-3.0 m, 3.0 m, -2.0 m), and the vector component form is (-3.0i + 3.0j - 2.0k) m. The magnitude of this displacement is approximately 4.69 m.

Step-by-step explanation:

To find the scalar components of the displacement vector, we subtract the coordinates of the initial position from those of the final position: De = e - b. This gives us:

• Dx = xe - xb = 1.0 m - 4.0 m = -3.0 m
• Dy = ye - yb = 4.5 m - 1.5 m = 3.0 m
• Dz = ze - zb = 0.5 m - 2.5 m = -2.0 m

So, the vector component form of the fly's displacement is D = (-3.0i + 3.0j - 2.0k) m.

To calculate the magnitude of the displacement vector, apply the formula for three-dimensional space:

D = \(\sqrt{Dx^2 + Dy^2 + Dz^2}\)

D = \(\sqrt{(-3.0)^2 + (3.0)^2 + (-2.0)^2}\) m

D = \(\sqrt{9 + 9 + 4}\) m

D = \(\sqrt{22}\) m

D ≈ 4.69 m

The correct option is not listed above, as the magnitude calculation based on the given scalar components would be approximately 4.69 m.