Let's break down your questions step by step, starting with the vector's magnitude and direction, then moving on to the houseflies and the vector components, and finally addressing the net displacement from your driving scenario.
Calculating the Magnitude of the Vector
To find the magnitude of a vector given its components, we can use the Pythagorean theorem. The formula for the magnitude \( M \) of a vector with components \( x \) and \( y \) is:
M = √(x² + y²)
In your case, the x component is 14 and the y component is -4. Plugging in these values:
- M = √(14² + (-4)²)
- M = √(196 + 16)
- M = √212
- M ≈ 14.6
So, the magnitude of the vector is approximately 14.6 units when rounded to the nearest tenth.
Finding the Direction of the Vector
The direction of the vector can be determined using the arctangent function, which gives us the angle \( θ \) relative to the positive x-axis:
θ = arctan(y/x)
Substituting the values:
- θ = arctan(-4/14)
- θ = arctan(-0.2857)
- θ ≈ -16.7 degrees
Thus, the direction of the vector is approximately -16.7 degrees, which falls within the range of -180° to 180° as required.
Estimating Houseflies on a Card
To determine how many common houseflies can fit on a 3-inch by 5-inch card, we first need to know the average size of a housefly. Typically, a housefly is about 0.25 inches long and 0.15 inches wide. For simplicity, we can approximate the area occupied by one fly.
Calculating the area of the card:
- Area of the card = 3 inches × 5 inches = 15 square inches
Now, estimating the area occupied by one fly (assuming it takes up a rectangular space):
- Area per fly ≈ 0.25 inches × 0.15 inches = 0.0375 square inches
Now, we can find out how many flies fit on the card:
- Number of flies = Area of card / Area per fly
- Number of flies = 15 / 0.0375 ≈ 400
So, approximately 400 houseflies can fit one layer deep on the card, rounded to one significant figure.
Finding Components of a Vector
For a vector with a magnitude of 10 and an angle of -60 degrees with the positive x-axis, we can find its components using trigonometric functions:
- x component = magnitude × cos(angle)
- y component = magnitude × sin(angle)
Calculating the components:
- x = 10 × cos(-60°) = 10 × 0.5 = 5
- y = 10 × sin(-60°) = 10 × (-√3/2) ≈ -8.7
Thus, the components of the vector are 5 and -8.7.
Calculating Net Displacement from Driving
When you drive west at 20 km/h for one hour, you cover a distance of 20 km west. Then, driving east at 15 km/h for one hour, you cover 15 km east. To find the net displacement, we can consider the distances in opposite directions:
- Displacement west = -20 km
- Displacement east = +15 km
Now, adding these displacements together:
- Net displacement = -20 km + 15 km = -5 km
This means your net displacement is 5 km to the west.
In summary, we calculated the vector's magnitude and direction, estimated how many houseflies fit on a card, found the components of a vector, and determined the net displacement from your driving scenario. Each step involved applying fundamental mathematical principles and reasoning through the problem logically.
