Physics 5054/11 — May/June 2016

Walkthrough

The question asks for the resultant of two vectors shown in Fig. 1. The parallelogram law states that if two vectors are drawn from a common point (tail-to-tail) as adjacent sides of a parallelogram, the resultant is the diagonal drawn from that common point to the opposite corner.
In Fig. 1, the two vectors are:

1. A horizontal vector pointing to the right.
2. A diagonal vector pointing upwards and to the right.
   Placing them tail-to-tail at the bottom-left, the parallelogram is completed with dashed lines. The resultant $R$ must start at the bottom-left (the common origin) and point to the top-right corner.
   Looking at the options:

• A: $R$ points from top-right to bottom-left (wrong direction).
• B: $R$ points downwards (incorrect construction).
• C: $R$ points from bottom-right to top-left (this represents the difference between the vectors, not the sum).
• D: $R$ points from bottom-left to top-right, correctly representing the sum of the two vectors.
  Therefore, D is the correct diagram.

Key Takeaways

• The resultant of two vectors can be found using the parallelogram law.
• The resultant vector is drawn from the common origin (tail) of the two vectors to the opposite corner of the parallelogram.
• The direction of the resultant is crucial; pointing it backwards gives the negative of the resultant.

Common Mistakes

• Choosing a diagram where $R$ points in the wrong direction (e.g., from the opposite corner back to the origin, as in A).
• Confusing vector addition (parallelogram/triangle method for sum) with vector subtraction (where $R$ would be the third side of the triangle pointing from the tip of one vector to the tip of the other, as in C).
• Not placing the vectors tail-to-tail before drawing the parallelogram.

Things to Be Careful About

• Vectors are quantities with both magnitude and direction. The direction of the resultant arrow must be correct.
• In the triangle method (tip-to-tail), the resultant is drawn from the tail of the first vector to the tip of the second vector. In the parallelogram method (tail-to-tail), it is drawn from the common tail to the opposite corner. Both give the same resultant.
• Ensure the dashed lines are drawn parallel to the original vectors to complete the parallelogram correctly.