Physics · Work, Energy And Power · NEET
Follow three steps. Step 1: find the dot product A.B = Ax Bx + Ay By + Az Bz (multiply matching components, then add). Step 2: find each magnitude, |A| = sqrt(Ax^2 + Ay^2 + Az^2) and the same for B. Step 3: put them in cos theta = (A.B)/(|A||B|) and take theta = cos inverse of that value. Example: A = 3i + 4j and B = 4i + 3j. A.B = 3(4) + 4(3) = 24. |A| = 5, |B| = 5. cos theta = 24/25 = 0.96, so theta = about 16.3 degrees.
The dot product is defined as A.B = |A| |B| cos theta. The size of the vectors is already baked into this value, so a big dot product might just mean the vectors are long, not that they point the same way. To get the pure angle information you must remove the lengths by dividing by |A| and |B|. What is left, (A.B)/(|A||B|), is exactly cos theta - a number between -1 and +1 that depends only on direction, not on how long the vectors are.
The sign is a quick angle check because |A| and |B| are always positive. If A.B is positive, cos theta is positive, so theta is acute (between 0 and 90 degrees). If A.B is zero, theta is exactly 90 degrees (the vectors are perpendicular). If A.B is negative, cos theta is negative, so theta is obtuse (between 90 and 180 degrees). In NEET this instantly tells you whether work is positive, zero, or negative without finding the full angle.
Yes. Inverse cosine only returns values from 0 to 180 degrees, which is exactly the range of the angle between two vectors (they cannot enclose more than a straight line). So there is no ambiguity like there is with sine. You never have to worry about a second answer - each pair of vectors has one angle between them, and cos inverse gives it directly.
Work is W = F.d = |F| |d| cos theta, where theta is the angle between force and displacement. If a question gives F and d in i, j, k form, the fastest route is often to compute the dot product directly for work. But if a question asks for the angle between the force and the path, you rearrange to cos theta = (F.d)/(|F||d|). This same idea decides positive, negative, and zero work.
Try the real previous-year questions from this chapter — each with the answer and a full solution.
cos theta = (A.B) / (|A| |B|), where A.B = Ax Bx + Ay By + Az Bz is the dot product and |A|, |B| are the magnitudes. Then theta = cos inverse of that value.
Two non-zero vectors are perpendicular when their dot product is zero: A.B = 0. That makes cos theta = 0, so theta = 90 degrees. This is a very common NEET shortcut.
No. The angle between two vectors is always between 0 and 180 degrees, and cos inverse only returns values in that range, so the answer is always valid and unique.
A negative value of cos theta means the angle is obtuse (more than 90 degrees), so the force has a component opposite to the displacement. This gives negative work, like friction slowing an object down.
Yes, especially in i, j, k form. You just multiply matching components, add, divide by the magnitudes, and take cos inverse. Drawing is only helpful to check whether your angle looks acute or obtuse.
