Dot Product and Vector Projection
For nonzero vectors A and B, A dot B = AB cos(theta). The scalar component of A along B is (A dot B)/|B|, while the vector projection is [(A dot B)/|B|^2]B.
Why this shows up in the exam
Testing perpendicularity · Finding work by a constant force · Extracting a component along an inclined direction
Learn the idea
The dot product measures how much one vector lies along another. Projection is the signed shadow of one arrow on a chosen direction. The dot product packages that shadow with the magnitude of the reference vector and becomes zero for perpendicular vectors.
🧠 Memory hook: Dot means along: zero dot means no along-component.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- A_vector dot B_vector = AB cos(theta) — scalar product and included angle
- comp_B(A) = (A dot B)/|B| — signed scalar component of A along B
- proj_B(A) = [(A dot B)/|B|²] B_vector — vector projection of A on B
How to approach it
- 1Write both vectors in the same basis
- 2Take the component-wise dot product
- 3Use the requested scalar, vector, or angle form
Common slip-ups that cost marks
- •Confusing scalar and vector projection
- •Dropping the sign of cos theta
- •Dividing by |B| instead of |B| squared for vector projection
🌟 That's the whole idea — you've got this. Try the practice set below; every question you attempt makes it stick a little harder.
Original chapter practice
Original questions for this chapter, not past-paper questions or an exact mapping to this individual concept.
A particle has initial speed 2 m/s and constant acceleration 3 m/s^2 for 4 s. What distance does it cover?
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