A vector 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.?

Question

Answer

Problem Statement

A vector 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. Explain in detail.

Solution

To solve this problem, we will use the vector addition formula:

C = A + B

where C is the resultant vector, A and B are the given vectors.

Step 1: Resolve the Vectors

Resolve the given vectors into their x and y components using the following formulas:

Ax = A cos θ

Ay = A sin θ

Bx = B cos θ

By = B sin θ

where θ is the angle the vector makes with the x-axis.

Using these formulas, we get:

Ax = 3 cos 20° = 2.828 m

Ay = 3 sin 20° = 0.998 m

Bx = 4 cos 110° = -1.305 m

By = 4 sin 110° = 3.648 m

Step 2: Add the Components

Add the x and y components of the vectors to get the x and y components of the resultant vector:

Cx = Ax + Bx

Cy = Ay + By

Substituting the values, we get:

Cx = 2.828 - 1.305 = 1.523 m

Cy = 0.998 + 3.648 = 4.646 m

Step 3: Find the Magnitude and Direction of the Resultant Vector

Using the components of the resultant vector, we can find its magnitude and direction:

C = √(Cx² + Cy²)

θ = tan⁻¹(Cy/Cx)

Substituting the values, we get:

C = √(1.523² + 4.646²) = 4.829 m

θ = tan⁻¹(4.646/1.523) = 72.69°

Step 4: Final Answer

Therefore, the magnitude of the resultant vector is 4.829 m and it makes an angle of 72.69° with the x-axis.

Answer

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