To find the horizontal component of the resultant vector $\mathbf{R}$ (which is the sum of vectors $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$), we need to break each vector into its horizontal (x) and vertical (y) components and then sum the horizontal components of all three vectors.

Vector $\mathbf{a}$:

• Magnitude: $60 \, \text{N}$
• Direction: 245° (measured counterclockwise from the positive x-axis)

The horizontal component of $\mathbf{a_x}$: $\mathbf{a_x} = 60 \, \text{N} \times \cos(245^\circ)$ $\mathbf{a_x} = 60 \, \text{N} \times \cos(245^\circ) = 60 \times (-0.4226) = -25.36 \, \text{N}$

Vector $\mathbf{b}$:

• Magnitude: $70 \, \text{N}$
• Direction: 75° north of east (which is equivalent to 75° measured counterclockwise from the positive x-axis)

The horizontal component of $\mathbf{b_x}$: $\mathbf{b_x} = 70 \, \text{N} \times \cos(75^\circ)$ $\mathbf{b_x} = 70 \, \text{N} \times \cos(75^\circ) = 70 \times 0.2588 = 18.12 \, \text{N}$

Vector $\mathbf{c}$:

• Magnitude: $95 \, \text{N}$
• Direction: 87° west of north (which is equivalent to 177° measured counterclockwise from the positive x-axis)

The horizontal component of $\mathbf{c_x}$: $\mathbf{c_x} = 95 \, \text{N} \times \cos(177^\circ)$ $\mathbf{c_x} = 95 \, \text{N} \times \cos(177^\circ) = 95 \times (-0.9986) = -94.87 \, \text{N}$

Horizontal component of the resultant vector $\mathbf{R_x}$:

Now, summing the horizontal components of vectors $\mathbf{a}$, $\mathbf{b}$, and $\mathbf{c}$:

$\mathbf{R_x} = \mathbf{a_x} + \mathbf{b_x} + \mathbf{c_x}$ $\mathbf{R_x} = -25.36 + 18.12 - 94.87 = -102.11 \, \text{N}$

Thus, the horizontal component of the resultant vector is $\mathbf{R_x} = -102.11 \, \text{N}$.

Would you like more details on how to find the vertical component or the magnitude of the resultant vector?

Related questions:

1. How do you find the vertical component of the resultant vector?
2. How do you calculate the magnitude of the resultant vector?
3. How can you find the direction of the resultant vector?
4. What are the physical interpretations of vector components?
5. How do trigonometric identities apply to vector component calculations?

Tip: Always visualize vectors on a coordinate plane to better understand their direction and components.