Vector Components and Unit Vectors
In Cartesian coordinates, A = A_x i_hat + A_y j_hat + A_z k_hat, its magnitude is the square root of the sum of squared components, and its direction is specified by component ratios or direction cosines.
Why this shows up in the exam
Converting map directions into displacements · Resolving launch velocity into horizontal and vertical parts · Representing forces and velocities in three dimensions
Learn the idea
A vector becomes calculable when resolved into signed components along chosen perpendicular axes. An arrow carries magnitude and direction together. Resolving it along axes replaces one geometric object by independent signed parts that can be recombined without losing direction.
🧠 Memory hook: Components are signed shadows of the arrow on the axes.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- A_vector = A_x i_hat + A_y j_hat + A_z k_hat — Cartesian component representation
- |A| = sqrt(A_x² + A_y² + A_z²) — magnitude from mutually perpendicular components
- A_hat = A_vector/|A| — unit vector along a nonzero vector
How to approach it
- 1Draw and label the positive axes
- 2Resolve with sine or cosine relative to the stated axis
- 3Recombine or normalize only after preserving signs
Common slip-ups that cost marks
- •Using unsigned component magnitudes
- •Measuring an angle from the wrong axis
- •Forgetting to normalize when a unit vector is requested
🌟 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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