Short answers to common vector questions.
1) Scalar vs vector in one sentence?
Scalar carries only magnitude. Vector carries magnitude and direction.
2) Why can’t I just add two forces arithmetically?
Because the directions may differ. 50 N north + 50 N east = 70.7 N northeast, not 100 N.
3) If I only have magnitude, what do I know?
Half the information. Real physics needs direction — which way, at what angle, in which quadrant.
4) Is a vector in a different location a different vector?
No. Position doesn’t matter. Same magnitude and direction = same vector. This freedom is what makes tip-to-tail addition work.
5) When to use graphical vs component method?
- Graphical: for initial intuition, small (2-3) vector problems, conceptual grasp
- Component: any real problem. More accurate, faster, scalable
6) If a vector is in quadrant II or III, how do I compute the angle?
\arctan(a_y/a_x) in a calculator only returns quadrants I and IV. If a_x < 0 (Q II or III), add 180°.
Better: identify the quadrant from signs of a_x and a_y, then use the correct reference angle.
7) Difference between \hat i, \hat j, \hat k and \hat x, \hat y, \hat z?
None. Both notations name the Cartesian unit vectors. \hat i, \hat j, \hat k is traditional; \hat x, \hat y, \hat z is more common in modern texts.
8) How to make a unit vector?
Divide a vector by its magnitude: \hat a = \vec a / |\vec a|. Result has magnitude 1 in the direction of \vec a.
9) Fundamental difference between dot and cross product?
- Dot: returns a scalar. Commutative. For projection and work
- Cross: returns a vector. Anti-commutative. For torque and rotation
10) When is the dot product zero?
When the angle between two vectors is 90° (perpendicular). Useful as a perpendicularity test.
11) When is the cross product zero?
When the vectors are parallel or antiparallel (\phi = 0° or 180°), or when either is zero.
12) Can I use my left hand for the right-hand rule?
No, direction flips. Always right hand even if left-handed. Or: look at the palm, thumb points to the answer.
13) Torque and work — why different formulas?
- Work:
W = \vec F \cdot \vec d— dot product, scalar - Torque:
\vec \tau = \vec r \times \vec F— cross product, vector
Because work is “amount of energy” (scalar) and torque is “tendency to rotate about an axis” (vector along that axis).
14) In 2D, what does the cross product look like?
Only the z component survives (since both vectors lie in the xy plane, the result is perpendicular to that plane). That is, (a_x b_y - a_y b_x)\hat k.
15) If two vectors are parallel, what are the operation results?
- Dot:
|a||b|(maximum) - Cross:
0 - Sum:
|a+b| = |a| + |b|(maximum)
16) How many components in a 3D vector?
Three: a_x, a_y, a_z. All three are needed for a complete description.
17) Why do vectors matter in physics but not accounting?
Because physics cares about directions. Momentum to the right and momentum to the left are physically different. In accounting, an amount is just an amount — no direction. (That said, vectors are used everywhere — AI, graphics, wherever “multiple related numbers” appear.)
18) Dot-product work vs absolute-value work — what’s the difference?
W = \vec F \cdot \vec d can be negative (when force opposes motion, like friction). |W| drops the sign, keeping only magnitude. Both useful in different contexts.
19) When can I not add two vectors?
If they represent different physical quantities (like velocity + force). Only vectors of the same units and dimension add.
20) How is a tensor different from a vector?
A vector is “multi-component” that transforms in a specific way under coordinate changes. A tensor generalizes vectors — can carry two or more directions. Example: inertia tensor of an object — how it accepts rotation. Needed in advanced mechanics.
Chapter wrap-up
Three key takeaways from Chapter 3:
- Vectors = the language of 3D physics. Without them, physics beyond 1D is confusing.
- Always use components. Graphical for intuition, component form for computation.
- Two kinds of vector product: dot (scalar → work), cross (vector → torque).
Next chapter: 2D motion — projectiles, circular motion, relative velocity. All built on vectors.
🎓 You’ve finished Chapter 3! Ready for 2D kinematics.
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