Physics · Motion In A Plane · NEET
It is always a scalar, meaning just a number with a unit but no direction. Even though you multiply two vectors, the cos theta term projects one vector onto the other, so the result loses direction. That is why it is called the scalar product. Remember: dot -> scalar, cross -> vector.
Work is done only by the part of the force that acts along the direction of motion. The dot product F.d = Fd cos theta automatically picks out that along-the-motion part (F cos theta) and multiplies it by the distance d. If the force is perpendicular to the motion (theta = 90 degrees), cos 90 = 0, so no work is done, which matches real physics like circular motion.
If A.B = 0 and neither vector is zero, then cos theta = 0, so theta = 90 degrees. The two vectors are perpendicular. This is a fast NEET test: to check if two vectors are at right angles, just compute Ax*Bx + Ay*By + Az*Bz and see if it equals zero.
The dot product uses cos: A.B = AB cos theta. The cross product uses sin. A simple way to keep them apart: the do-T product and cos-Theta both feel 'flat/along', while sin (cross) gives the perpendicular area. Mixing them up is the most common NEET error here.
Multiply matching components and add: if A = Ax i + Ay j + Az k and B = Bx i + By j + Bz k, then A.B = Ax*Bx + Ay*By + Az*Bz. This works because i.i = j.j = k.k = 1 and i.j = j.k = k.i = 0. No angle needed.
A.A = A*A*cos 0 = A^2, the square of its magnitude. So the magnitude of a vector is A = sqrt(A.A) = sqrt(Ax^2 + Ay^2 + Az^2). This link between dot product and magnitude is used a lot in energy problems.
A particle moves from a point (-2 i + 5 j) to (4 j + 3 k) when a force of (4 i + 3 j) N is applied. How much work is done by the force?
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.
Yes. A.B = B.A because AB cos theta does not depend on the order. This is different from the cross product, which changes sign when you swap the order.
When force (newton) is dotted with displacement (metre), the result W = F.d has the unit newton-metre, which is the joule (J), the unit of energy.
Yes. If theta is between 90 and 180 degrees, cos theta is negative, so A.B is negative. In work, a negative dot product means the force opposes the motion, like friction doing negative work.
i.j = 0 because they are perpendicular (cos 90 = 0). But i.i = 1 because a unit vector dotted with itself gives cos 0 = 1.
Rearrange the formula: cos theta = (A.B) / (AB). Compute A.B from components, divide by the product of the magnitudes, then take the inverse cosine to get theta.
