Vectors
The mathematical language for quantities that carry a direction
- The difference between a scalar and a vector.
- How to add and subtract vectors geometrically (tip-to-tail).
- How to resolve a vector into components and rebuild it.
- Unit-vector notation and adding vectors component by component.
- The two products: the dot product (a scalar) and the cross product (a vector).
What Is Physics?
Many physical quantities carry both a size and a direction — and to handle them we need a special language: the language of vectors. You already use it informally: "go five blocks down this street, then turn left" is a vector instruction. Physics and engineering lean on vectors constantly, especially once we reach rotation and magnetic forces.
Vectors vs Scalars
Has magnitude and direction and follows special rules of combination. Examples: displacement, velocity, acceleration, force. Drawn as an arrow.
Has magnitude only and obeys ordinary algebra. Examples: temperature, mass, time, energy. A signed number with a unit is enough.
The simplest vector is displacement — a change of position drawn as an arrow from start to finish. A displacement vector says nothing about the path taken; only the overall change from beginning to end. And a vector can be slid around the page freely: as long as its length and direction don't change, it's the same vector.
Adding Vectors Geometrically
To add two vectors by hand, use the tip-to-tail rule: draw the first to scale, draw the second starting at the head of the first, and the sum (or resultant) runs from the tail of the first to the head of the second.
Vector addition obeys two familiar laws:
Subtraction is just adding the reverse. The vector \(-b\) has the same length as \(b\) but points the opposite way, so:
You can add two displacements or two velocities, but adding a displacement to a velocity is meaningless — like trying to add 21 s to 12 m.
Components of a Vector
Drawing arrows is tidy but tedious. The powerful method is algebraic: drop a vector onto a set of axes and work with its components — its projections on the x and y axes. Finding them is called resolving the vector.
Going the other way — from components back to magnitude and direction:
A calculator's \(\tan ^{-1}\) only returns angles in two quadrants. Always sketch the vector: if the components place it in a different quadrant than the calculator's answer, add 180° to fix it. The right answer is the one that matches your drawing.
Unit Vectors
A unit vector has magnitude exactly 1 and no units — its only job is to point. The unit vectors along the x, y, and z axes are written \(\hat{\imath}\), \(\hat{\jmath}\), and \(\hat{k}\) (the "hat" marks them as unit vectors). With them, any vector becomes a clean sum:
Adding Vectors by Components
This is the everyday workhorse. To add vectors, just add their components axis by axis:
- Resolve every vector into its x, y (and z) components.
- Add the components separately, axis by axis, to get the components of the sum.
- Recombine — either leave the answer in unit-vector form, or convert to magnitude-and-angle.
Vectors & the Laws of Physics
We're free to orient the axes however we like. Rotate them, and a vector's components change — but its magnitude and direction do not. Every choice of axes describes the same vector.
That coordinate-independence is exactly why the laws of physics are written as vector equations. One compact equation like \(r = a + b\) stands in for three component equations at once.
Multiplying Vectors
There are three distinct "multiplications," and none works quite like ordinary arithmetic.
1 · Vector times a scalar
Multiplying a vector by a scalar \(s\) scales its length by \(|s|\). The direction is unchanged if \(s \gt 0\) and reversed if \(s \lt 0\).
2 · The scalar (dot) product → gives a number
Think of the dot product as "how much of one vector lies along the other." You'll meet it first as work (force along displacement) in Chapter 7.
3 · The vector (cross) product → gives a new vector
Point your right-hand fingers along \(a\) and curl them toward \(b\) through the smaller angle; your thumb gives the direction of \(a \times b\). Order matters: \(b \times a = -(a \times b)\). You'll meet the cross product as torque in Chapter 11.
Putting It to Work
Problem. A plane is sighted 215 km away, in a direction 22° east of due north. How far east and how far north is it?
Set up. Take +x east and +y north. Measured from the +x axis, the angle is \(\theta = 90^{\circ} - 22^{\circ} = 68^{\circ}\).
So the plane is about 81 km east and 2.0 × 10² km north of the airport.
Problem. Add \(a = 4.2\hat{\imath} - 1.5\hat{\jmath}\), \(b = -1.6\hat{\imath} + 2.9\hat{\jmath}\), and \(c = -3.7\hat{\jmath}\) (metres).
As a magnitude and angle: \(r = \sqrt{2.6^{2} + 2.3^{2}} \approx 3.5 m\) at \(\tan ^{-1}(-2.3 / 2.6) \approx -41^{\circ}\) (i.e. 41° clockwise from +x).
Problem. Find the angle between \(a = 3.0\hat{\imath} - 4.0\hat{\jmath}\) and \(b = -2.0\hat{\imath} + 3.0\hat{k}\).
Solution. Magnitudes: \(a = \sqrt{3^{2} + 4^{2}} = 5.00\), \(b = \sqrt{2^{2} + 3^{2}} = 3.61\). Component dot product: \(a \cdot b = (3)(-2) + (-4)(0) + (0)(3) = -6.0\).
The negative dot product already told us the angle is obtuse (more than 90°).
Chapter Summary
Scalars have magnitude only; vectors have magnitude and direction and obey vector algebra.
Tip-to-tail. Addition is commutative and associative; subtract by adding the reverse vector.
\(a_{x} = a \cos \theta\), \(a_{y} = a \sin \theta\); rebuild with \(a = \sqrt{a_{x}^{2} + a_{y}^{2}}\), \(\tan \theta = a_{y}/a_{x}\).
\(a = a_{x}\hat{\imath} + a_{y}\hat{\jmath} + a_{z}\hat{k}\). Add vectors component by component.
\(a \cdot b = ab \cos \varphi\) → a scalar; zero when perpendicular.
\(|a \times b| = ab \sin \varphi\) → a vector ⟂ to both, by the right-hand rule; zero when parallel.
Problems
Sketch each situation first — a quick diagram catches most sign and quadrant errors.
- A vector in the xy plane has magnitude 7.3 m and points 250° counterclockwise from the +x axis. Find its (a) x and (b) y components.
- Vector A has \(A_{x} = -25.0 m\) and \(A_{y} = 40.0 m\). Find (a) its magnitude and (b) the angle it makes with the +x axis.
- A ship meant to sail 120 km due north is blown to a point 100 km due east of its start. (a) How far and (b) in what direction must it now sail to reach its destination?
- A person walks 3.1 km north, 2.4 km west, then 5.2 km south. (a) How far and (b) in what direction is the final point from the start?
- Add \(a = 4.0\hat{\imath} + 3.0\hat{\jmath}\) and \(b = -13.0\hat{\imath} + 7.0\hat{\jmath}\) (metres). Give the sum's (a) unit-vector form, (b) magnitude, and (c) direction.
- A car drives 50 km east, 30 km north, then 25 km at 30° east of north. Find the (a) magnitude and (b) angle of the total displacement.
- For \(a = 3.0\hat{\imath} + 5.0\hat{\jmath}\) and \(b = 2.0\hat{\imath} + 4.0\hat{\jmath}\), compute (a) \(a \times b\) and (b) \(a \cdot b\).
- Vectors C and D have magnitudes 3 and 4 units. Find the angle between them if \(C \cdot D\) equals (a) 0, (b) +12, and (c) −12 units.
- Vector A has magnitude 6.00 and B has magnitude 7.00, with \(A \cdot B = 14.0\). What is the angle between them?
- Given \(a = 3.0\hat{\imath} - 4.0\hat{\jmath}\), find (a) the magnitude and (b) the direction of \(a\), then (c) the magnitude and (d) direction of \(-2.0a\).