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Vectors

Source lecture(s): SC133 Lec 3

Intuition

"Walk 5 km" is incomplete; "walk 5 km north-east" is an instruction. Quantities that carry both magnitude and direction — displacement, velocity, force — are vectors. Quantities fully specified by one number — mass, temperature, speed — are scalars. The entire machinery of 2-D and 3-D physics is bookkeeping for arrows.

Components: the master move

Any vector resolves into perpendicular components:

\[\vec A = A_x\,\hat\imath + A_y\,\hat\jmath, \qquad A_x = A\cos\theta,\quad A_y = A\sin\theta\]
\[A = |\vec A| = \sqrt{A_x^2 + A_y^2}, \qquad \tan\theta = \frac{A_y}{A_x}\]

Components turn geometry into arithmetic: to add vectors, add their components. \(\vec C = \vec A + \vec B \iff C_x = A_x + B_x,\ C_y = A_y + B_y\). Physics problems in 2-D are just two 1-D problems glued together — the insight behind projectile motion.

The two products

Dot product (scalar result) — measures alignment:

\[\vec A\cdot\vec B = AB\cos\theta = A_xB_x + A_yB_y + A_zB_z\]

Zero for perpendicular vectors. Physics use: work \(W = \vec F\cdot\vec d\) — only the force component along the motion counts.

Cross product (vector result) — measures perpendicularity, points normal to both (right-hand rule):

\[|\vec A\times\vec B| = AB\sin\theta, \qquad \vec A\times\vec B = (A_yB_z - A_zB_y)\,\hat\imath + (A_zB_x - A_xB_z)\,\hat\jmath + (A_xB_y - A_yB_x)\,\hat k\]

Physics use: torque \(\vec\tau = \vec r\times\vec F\), angular momentum \(\vec L = \vec r\times\vec p\), magnetic force (SC134).

Worked example: river crossing

A boat moves at \(4\,\text{m/s}\) relative to water, pointed straight across a river flowing at \(3\,\text{m/s}\). What is the boat's speed over the ground?

The velocities add as vectors at right angles: \(v = \sqrt{4^2 + 3^2} = 5\,\text{m/s}\), at \(\theta = \tan^{-1}(3/4) \approx 37°\) downstream of straight across. (This is relative motion in one picture.)

Common mistakes

  • Adding magnitudes instead of components. \(|\vec A + \vec B| \ne A + B\) unless the vectors are parallel.
  • Dropping direction from an answer. "The force is 10 N" is half an answer.
  • Using degrees in one place and radians in another. Pick one; check your calculator mode.
  • Right-hand-rule with the left hand. It happens to everyone once. Only once, ideally.

Knowledge graph position

Prerequisites: Measurement. Leads to: Projectile motion, relative motion, Newton's laws, torque.

Quiz

Q1 (computational). \(\vec A = 3\hat\imath + 4\hat\jmath\), \(\vec B = -2\hat\imath + \hat\jmath\). Find \(\vec A\cdot\vec B\) and \(|\vec A\times\vec B|\).

Answer

\(\vec A\cdot\vec B = -6 + 4 = -2\). \(\vec A\times\vec B = (3\cdot1 - 4\cdot(-2))\hat k = 11\hat k\), magnitude 11.

Q2 (conceptual). Can the magnitude of \(\vec A + \vec B\) be smaller than both \(A\) and \(B\)?

Answer

Yes — nearly antiparallel vectors almost cancel: \(|\vec A + \vec B|\) ranges from \(|A - B|\) to \(A + B\).

Q3 (multiple choice). \(\vec A\cdot(\vec A\times\vec B)\) equals: (a) \(A^2B\) (b) 0 (c) \(AB\cos\theta\)

Answer

(b). The cross product is perpendicular to \(\vec A\), and dotting perpendicular vectors gives zero — always.