Let's break down your questions step by step, starting with the first one about the car's journey. This involves both graphical representation and calculation of displacement. We'll then move on to the vector operations and other queries you've posed.
1. Car's Journey Analysis
a) Graphical Representation of Displacement
To represent the car's journey graphically, we can use a coordinate system. The car travels:
- 3 km due north (which we can represent as moving up on the y-axis)
- 4 km east (which we can represent as moving right on the x-axis)
Starting from the origin (0,0), the car moves to the point (0,3) after traveling north. Then, moving east, it reaches the point (4,3). You can visualize this as a right triangle where:
- The vertical leg is 3 km (north)
- The horizontal leg is 4 km (east)
b) Resultant Displacement Calculation
The resultant displacement can be found using the Pythagorean theorem, since the path forms a right triangle:
Resultant Displacement (R) = √(3² + 4²) = √(9 + 16) = √25 = 5 km
The direction of this displacement can be found using trigonometry. The angle θ with respect to the north can be calculated as:
tan(θ) = opposite/adjacent = 4/3
Thus, θ = arctan(4/3), which gives you the angle east of north.
2. Vector Operations
Given Vectors
Let’s define the vectors:
- A = (1, 4)
- B = (-3, 5)
- C = (2, 9)
a) A + B
To add vectors A and B, we simply add their corresponding components:
A + B = (1 + (-3), 4 + 5) = (-2, 9)
b) A - C
Now, for A - C:
A - C = (1 - 2, 4 - 9) = (-1, -5)
c) A + B + C
For the sum of all three vectors:
A + B + C = (-2 + 2, 9 + 9) = (0, 18)
d) Interpretation of Result
The result (0, 18) means that the net effect of these vectors is a displacement of 18 units in the positive y-direction, with no displacement in the x-direction. This indicates that the vectors balance out in the horizontal direction.
3. Airplane Displacement
a) Final Displacement Calculation
The airplane flies to point A and then to point B:
- From the origin to A: (1500, 3000, 200)
- From A to B: (200, 5000, 100)
The total displacement from the origin to B is:
Final Displacement = (1500 + 200, 3000 + 5000, 200 + 100) = (1700, 8000, 300)
b) Magnitude of the Displacement
The magnitude of the displacement can be calculated using the 3D distance formula:
Magnitude = √(1700² + 8000² + 300²)
Calculating this gives:
Magnitude = √(2890000 + 64000000 + 90000) = √(66890000) ≈ 2580.5 m
4. Adding Displacements
Combining Two Displacements
We have:
- A = 6 km due north
- B = 4 km at an angle of 30 degrees above the horizontal
To add these, we need to break down B into its components:
- B_x = 4 * cos(30°) = 4 * (√3/2) ≈ 3.46 km
- B_y = 4 * sin(30°) = 4 * (1/2) = 2 km
Now, adding the components:
- Total x = 0 + 3.46 = 3.46 km
- Total y = 6 + 2 = 8 km
The resultant displacement is approximately (3.46, 8) km.
5. Vector Concepts
a) Null Vector
A null vector, also known as a zero vector, is a vector that has a magnitude of zero and no specific direction. It is represented as (0, 0) in two dimensions. It plays a crucial role in vector addition, as adding a null vector to any vector does not change the original vector.
b) Equivalent Vectors
Equivalent vectors are vectors that have the same magnitude and direction, regardless of their initial points. For example, two vectors of 5 units pointing north are equivalent, even if they start from different locations.
6. Orthogonality of Vectors
Checking Orthogonality
Given the displacements:
To determine if they are orthogonal, we check the dot product:
A · B = (3 * 10) + (5 * -6) = 30 - 30 = 0
Since the dot product is zero, vectors A and B are orthogonal, meaning they are at right angles to each other.
Feel free to ask if you have any further questions or need clarification on any of these topics!
