1. Find the horizontal and vertical component of the vector: 12
m, 25 ̊south of west.
2. Vector A of a magnitude 5.0 units has a horizontal component
of 2.0 units. (a) Find the angle that vector A makes with the
horizontal. Also (b) determine the vertical component of
vector A
1. To find the horizontal and vertical components of a vector that is given in magnitude and direction:
Step 1: Identify the given information:
- Magnitude: 12 m
- Direction: 25° south of west
Step 2: Draw a diagram:
Start by drawing a coordinate system with the x-axis representing the east-west direction and the y-axis representing the north-south direction. Place the vector tail at the origin.
Step 3: Find the horizontal component:
Since the vector is south of west, the horizontal component will be negative. To find it, use trigonometry. Draw a right triangle using the given information, with the angle of 25°. The adjacent side represents the horizontal component.
Horizontal component = Magnitude * cos(angle)
Horizontal component = 12 m * cos(25°)
Step 4: Find the vertical component:
Since the vector is south of west, the vertical component will also be negative. To find it, use trigonometry. Draw a right triangle using the given information, with the angle of 25°. The opposite side represents the vertical component.
Vertical component = Magnitude * sin(angle)
Vertical component = 12 m * sin(25°)
So, the horizontal component is 12 m * cos(25°) and the vertical component is -12 m * sin(25°).
2. To find the angle and the vertical component of a vector given its horizontal component and magnitude:
Step 1: Identify the given information:
- Magnitude: 5.0 units
- Horizontal component: 2.0 units
Step 2: Draw a diagram:
Start by drawing a coordinate system with the x-axis representing the horizontal direction and the y-axis representing the vertical direction. Place the vector tail at the origin.
Step 3: Find the angle:
To find the angle that vector A makes with the horizontal, use trigonometry. Draw a right triangle using the given information, with the adjacent side representing the horizontal component.
Angle = arccos(horizontal component / magnitude)
Angle = arccos(2.0 / 5.0)
Step 4: Determine the vertical component:
To find the vertical component, use trigonometry. Draw a right triangle using the given information. The opposite side represents the vertical component.
Vertical component = magnitude * sin(angle)
Vertical component = 5.0 * sin(arccos(2.0 / 5.0))
So, the angle that vector A makes with the horizontal is arccos(2.0 / 5.0) radians (or degrees). And the vertical component of vector A is 5.0 * sin(arccos(2.0 / 5.0)).