Physics · Motion In A Plane · NEET
Only if the two vectors point in the exact same direction (theta = 0). In general you must use R = sqrt(A^2 + B^2 + 2AB cos theta). Simple addition A + B ignores direction and gives a wrong, too-large answer. For example, two forces of 3 N and 4 N at 90 degrees give R = 5 N, not 7 N.
Theta is the angle BETWEEN the two vectors when they are drawn from the same starting point (tail to tail). It is NOT the angle either vector makes with the horizontal. A very common mistake in NEET is plugging in the angle with the x-axis. Always place both tails together and read the angle between them.
The cos theta comes from the cross-term when you square (A + B) using components. The magnitude of the sum depends on how much the two vectors point the SAME way, and that overlap is measured by cos theta. Sin theta appears only in the DIRECTION formula (tan alpha = B sin theta / (A + B cos theta)), not in the magnitude.
Maximum when theta = 0 (vectors parallel): R_max = A + B, because cos 0 = 1. Minimum when theta = 180 degrees (vectors opposite): R_min = |A - B|, because cos 180 = -1. So the resultant of two vectors always lies between |A - B| and A + B. NEET loves asking which R value is possible for given A and B.
Use tan alpha = (B sin theta) / (A + B cos theta), where alpha is the angle the resultant makes with vector A. Take arctan to get alpha. Magnitude (how long) and direction (which way) are two separate calculations - NEET numericals often ask for both.
Try the real previous-year questions from this chapter — each with the answer and a full solution.
Magnitude: R = sqrt(A^2 + B^2 + 2AB cos theta). Direction (angle alpha with A): tan alpha = (B sin theta) / (A + B cos theta). Here theta is the angle between the two vectors.
Maximum = A + B (when theta = 0, vectors parallel). Minimum = |A - B| (when theta = 180 degrees, vectors opposite). Any resultant of the two must lie between these two values.
When theta = 90 degrees, cos 90 = 0, so R = sqrt(A^2 + B^2). For example, 3 and 4 at right angles give sqrt(9 + 16) = 5. The direction is tan alpha = B/A.
Yes. The formula R = sqrt(A^2 + B^2 + 2AB cos theta) is derived directly from the parallelogram law of vector addition, where R is the length of the diagonal of the parallelogram formed by A and B.
No. Vector addition is commutative, so A + B = B + A. The magnitude and direction of the resultant are the same whichever vector you call A. Only the reference angle alpha changes depending on which vector you measure from.
