1. A vecor A makes an angle of 20 degree and vector B makes an angle of 110 degree with the X-axis. The magnitudes of these vectors are 3m and 4m respectively. Find the resultant

2. A spy report about a suspected car read as follows. The car moved 2.00 km east, made a perpendicular left turn, ran for 500m, made a perpendicular right turn, ran for 4.00 km and stopped. Find the displacement of car.

Kindly explain these two problems with proper explanation and diagram

Let's tackle these two problems step by step, starting with the vector addition and then moving on to the displacement of the car. Both problems involve some fundamental concepts in physics and mathematics, particularly in vector analysis.

Finding the Resultant of Two Vectors

We have two vectors, A and B, with the following characteristics:

• Vector A: Magnitude = 3 m, Angle = 20°
• Vector B: Magnitude = 4 m, Angle = 110°

To find the resultant vector, we can break each vector into its components along the X and Y axes. The components can be calculated using trigonometric functions:

Step 1: Calculate Components of Vector A

For vector A:

• X-component: A_x = A * cos(θ) = 3 * cos(20°)
• Y-component: A_y = A * sin(θ) = 3 * sin(20°)

Calculating these values:

• A_x ≈ 3 * 0.9397 ≈ 2.819 m
• A_y ≈ 3 * 0.3420 ≈ 1.026 m

Step 2: Calculate Components of Vector B

For vector B:

• X-component: B_x = B * cos(θ) = 4 * cos(110°)
• Y-component: B_y = B * sin(θ) = 4 * sin(110°)

Calculating these values:

• B_x ≈ 4 * (-0.3420) ≈ -1.368 m
• B_y ≈ 4 * 0.9397 ≈ 3.759 m

Step 3: Sum the Components

Now, we can find the resultant vector components:

• R_x = A_x + B_x = 2.819 - 1.368 ≈ 1.451 m
• R_y = A_y + B_y = 1.026 + 3.759 ≈ 4.785 m

Step 4: Calculate the Magnitude and Direction of the Resultant

The magnitude of the resultant vector R can be found using the Pythagorean theorem:

• |R| = √(R_x² + R_y²)

Calculating this gives:

• |R| = √(1.451² + 4.785²) ≈ √(2.107 + 22.911) ≈ √(25.018) ≈ 5.0 m

The direction (angle θ) of the resultant vector can be found using the tangent function:

• θ = tan^-1(R_y/R_x) = tan^-1(4.785/1.451)

Calculating this gives:

• θ ≈ 73.74°

Thus, the resultant vector R has a magnitude of approximately 5.0 m and makes an angle of about 73.74° with the X-axis.

Calculating the Displacement of the Car

Now let's analyze the movement of the car based on the spy report:

• Moves 2.00 km east
• Makes a left turn (now facing north) and runs for 500 m
• Makes a right turn (now facing east) and runs for 4.00 km

Step 1: Break Down the Movements

We can visualize the car's path as follows:

• First leg: 2.00 km east (let's call this point A)
• Second leg: 500 m north (let's call this point B)
• Third leg: 4.00 km east (let's call this point C)

Step 2: Calculate the Total Displacement

The total eastward movement is:

• 2.00 km + 4.00 km = 6.00 km

The northward movement is simply 500 m, which is equivalent to 0.5 km.

Step 3: Use the Pythagorean Theorem

Now we can find the resultant displacement using the Pythagorean theorem:

• Displacement = √(eastward movement² + northward movement²)

Calculating this gives:

• Displacement = √(6.00² + 0.5²) = √(36 + 0.25) = √(36.25) ≈ 6.02 km

The direction of the displacement can be found using the tangent function:

• θ = tan^-1(northward movement/eastward movement) = tan^-1(0.5/6.00)

Calculating this gives:

• θ ≈ 4.76° north of east

In summary, the displacement of the car is approximately 6.02 km at an angle of about 4.76° north of east.