Question

Parallelogram law of vectors

Answer 1

[AI-generated answer — Step-by-Step Bot]

The parallelogram law of vectors states that if two vectors are represented by the sides of a parallelogram with a common starting point, then the resultant vector is represented by the diagonal of the parallelogram passing through that common starting point.

To illustrate this, suppose we have two vectors, vector A and vector B, with the same starting point. We can represent vector A by one side of the parallelogram and vector B by another side. The remaining sides of the parallelogram will represent the negative of each vector (-A and -B).

Now, if we connect the common starting point with the opposite vertex of the parallelogram, that diagonal represents the resultant vector (vector R). Vector R is the sum of vector A and vector B.

Mathematically, the parallelogram law can be stated as:

R = A + B

where R is the resultant vector, A is the first vector, and B is the second vector.

This law is based on the concept of vector addition, which involves adding the corresponding components of the vectors together. By applying the parallelogram law, we can determine the magnitude and direction of the resultant vector in a step-by-step manner.

Answer 2

[AI-generated answer — Bot GPT 3.5]

The parallelogram law of vectors states that if two vectors acting simultaneously at a point can be represented in magnitude and direction by two adjacent sides of a parallelogram drawn from a point, then their resultant vector is represented in magnitude and direction by the diagonal of the parallelogram originating from the same point.

Mathematically, if we have two vectors A and B acting at a point P, then the resultant vector R can be found using the parallelogram law:

R = A + B

To visually represent this law, we draw the vectors A and B as sides of a parallelogram with P as one of the vertices. The resultant vector R is represented by the diagonal of the parallelogram originating from P.

The magnitude of the resultant vector R can be found using the law of cosines:

|R| = sqrt(|A|^2 + |B|^2 + 2|A||B|cosθ)

where |A| and |B| are the magnitudes of vectors A and B, and θ is the angle between them.

The direction of the resultant vector R can be found using the law of sines:

sin(α) / |A| = sin(θ) / |R|

where α is the angle between the resultant vector R and vector A.

The parallelogram law of vectors is widely used in physics and engineering to find the net effect of multiple forces acting on an object or the result of adding multiple velocity vectors.

Answer 3

[AI-generated answer — Explain Bot]

The parallelogram law of vectors is a fundamental principle in vector addition. It states that if two vectors are represented by the adjacent sides of a parallelogram, then the diagonal of the parallelogram, which connects the opposite corners, represents the resultant vector obtained by adding the two vectors.

Mathematically, if two vectors A and B are represented by the sides of a parallelogram, then the resultant vector R can be found by connecting the opposite corners of the parallelogram:

R = A + B

To better understand the concept, let's go through a step-by-step explanation of how to apply the parallelogram law of vectors:

Step 1: Represent the vectors A and B graphically as directed line segments, starting from a common initial point.

Step 2: Complete the parallelogram by drawing the second side of the parallelogram parallel to the vector B starting from the terminal point of vector A.

Step 3: Draw the diagonal of the parallelogram, which connects the initial point of vector B to the terminal point of vector A. This diagonal represents the resultant vector R.

Step 4: Measure the magnitude (length) and direction of the diagonal vector to determine the magnitude and direction of the resultant vector.

It's important to note that this principle holds true regardless of the length or direction of the vectors involved. The parallelogram law of vectors is a geometric method that is applicable to both two-dimensional and three-dimensional vectors.

By applying the parallelogram law of vectors, you can accurately determine the resultant vector when adding multiple vectors together.