Physics · Motion In A Plane · NEET
No, they give the exact same resultant. In the triangle law you place the tail of the second vector at the head of the first (head-to-tail). In the parallelogram law you place both vectors tail-to-tail (starting from the same point) and complete the parallelogram. Both are just two ways to draw the same addition, so R = sqrt(A^2 + B^2 + 2AB cos theta) works for either. Use parallelogram when two forces or velocities act at the same point.
In the parallelogram, theta is the angle between the two vectors when they are drawn from the same starting point (tail-to-tail). When you drop a perpendicular to derive R, the projection A + B cos theta adds up, giving a plus sign. The cosine rule of a triangle has a minus sign because there the angle used is the interior angle (180 - theta) of the triangle, and cos(180 - theta) = -cos theta. Same physics, just a different angle labelled.
Theta is the angle between the two vectors A and B when both are drawn starting from the same point. It is NOT the angle of the diagonal. If two forces point in the same direction, theta = 0 (maximum resultant). If they are perpendicular, theta = 90 and cos theta = 0, so R = sqrt(A^2 + B^2). If they are opposite, theta = 180 (minimum resultant).
Size alone is not enough in NEET. The angle beta that the resultant makes with vector A is given by tan(beta) = B sin theta / (A + B cos theta). For example, with an 8 N and a 6 N force at 90 degrees, tan(beta) = 6/8 = 3/4 measured from the 8 N force. Always report both magnitude and direction.
Resultant is maximum R = A + B when theta = 0 (vectors along the same line, same direction). It is minimum R = |A - B| when theta = 180 (opposite directions). So the resultant of two vectors always lies between |A - B| and A + B. This range is a very common NEET trap.
The magnitude and direction of the acceleration produced in a 5 kg body by two forces of 8 N and 6 N acting perpendicular to each other are:
If the magnitude of the sum of two vectors is equal to the magnitude of the difference of the two vectors, the angle between these vectors is:
Try the real previous-year questions from this chapter — each with the answer and a full solution.
If two vectors acting simultaneously at a point are represented in magnitude and direction by the two adjacent sides of a parallelogram drawn from that point, then their resultant is represented in magnitude and direction by the diagonal of the parallelogram drawn from the same point.
R = sqrt(A^2 + B^2 + 2AB cos theta), where A and B are the magnitudes of the two vectors and theta is the angle between them. When theta = 90, this simplifies to R = sqrt(A^2 + B^2).
The resultant makes an angle beta with vector A given by tan(beta) = B sin theta / (A + B cos theta). Both the magnitude R and this angle beta are needed for a complete answer in NEET.
Adding two forces, two velocities, or two momenta at a point appears every year in Motion in a Plane and Laws of Motion. Perpendicular cases (giving neat 3-4-5 or 5-12-13 style answers) are the most common, so the parallelogram formula is a fast, guaranteed tool.
Maximum is A + B (when theta = 0) and minimum is |A - B| (when theta = 180). Any resultant of two vectors must lie between these two values.
