Linear Algebra and Analytic Geometry · 120 h · Topic 4 of 9

Vectors in R² and R³

Forces, fields, velocities and phasors have something in common: one number is not enough to describe them. You need a direction, and with it comes an algebra of its own.

Components Dot product Cross product Projection Triple product

01When one number is not enough

The temperature of a room, the resistance of a resistor or the energy of a battery can be described with a number and a unit: they are scalars. The force on an electron, the electric field at a point or the velocity of a moving object also need a direction and a sense: they are vectors.

A vector is drawn as an arrow, but what matters is its set of three data: magnitude, direction and sense. Two arrows drawn in different places represent the same vector if they have the same magnitude, the same direction and the same sense; that is why they are called free vectors.

Two products, not one

Between vectors there are two different products, and they must not be confused: the scalar (or dot) product, which gives a number, and the vector (or cross) product, which gives another vector. Each one answers a different physical question: how much one vector goes along with the other, and how perpendicular they are.

02Components, magnitude and unit vectors

In a set of perpendicular axes, a vector is written with its projections onto each axis:

\[ \vec{v} = (v_x, v_y, v_z) = v_x \hat{i} + v_y \hat{j} + v_z \hat{k} \quad \left| \vec{v} \right| = \sqrt{v_x^2 + v_y^2 + v_z^2} \] The magnitude is the Pythagorean theorem in three dimensions. The unit vectors î, ĵ and k̂ have magnitude 1 and point along the axes.

The unit vector of a vector is that same vector divided by its magnitude: it keeps the direction and the sense, and has length 1. It is used to separate “how much” from “which way”, which is one of the most useful operations in vector calculus.

Example

For v = (3, −4, 0): |v| = √(9 + 16) = 5, and its unit vector is (0.6, −0.8, 0). Check: 0.6² + 0.8² = 0.36 + 0.64 = 1. ✔

03Addition and scalar multiplication

They are added component by component, which is the same as the parallelogram rule: draw the two vectors from a common origin and take the diagonal. Multiplying by a scalar k stretches the vector by a factor of k, and if k is negative it reverses the sense.

A linear combination mixes the two operations: a·u + b·v. It is the idea that is later generalized in vector spaces, and it is also what a circuit does when it superposes the response to two sources.

04Dot product: how much one goes along with the other

\[ \vec{u} \cdot \vec{v} = u_x v_x + u_y v_y + u_z v_z = \left| \vec{u} \right| \left| \vec{v} \right| \cos \theta \] Two formulas for the same thing: one is computed from the components and the other explains what it means. Equating them gives the angle between the vectors.

It is zero if and only if the vectors are perpendicular (with neither one zero). It is the quickest way to check orthogonality.
It is positive if the angle is acute and negative if it is obtuse.
It is commutative and distributive, and u · u = |u|².

The projection of u onto v —the “shadow” of u in the direction of v— comes straight out of it:

\[ {\operatorname{proj}}_v \vec{u} = \frac{\vec{u} \cdot \vec{v}}{{\left| \vec{v} \right|}^2} \vec{v} \]

In physics, the work done by a force is W = F · d: only the component of the force that goes along with the displacement does work. In alternating current, the active power is P = |V||I| cos φ, which is exactly the dot product between the voltage and current phasors.

05Cross product: how perpendicular they are

It exists only in three-dimensional space. It gives a vector perpendicular to the plane of the two, with the sense given by the right-hand rule, and it is computed as a symbolic determinant:

\[ \vec{u} \times \vec{v} = \left| \begin{array}{ccc} \hat{i} & \hat{j} & \hat{k} \\ u_x & u_y & u_z \\ v_x & v_y & v_z \end{array} \right| \quad \left| \vec{u} \times \vec{v} \right| = \left| \vec{u} \right| \left| \vec{v} \right| \operatorname{sin} \theta \] Its magnitude is the area of the parallelogram they form. It is zero if they are parallel: then there is no parallelogram.

It is not commutative

u × v = −(v × u): reversing the order reverses the sense of the result. Nor is it associative. It is the operation that explains why the force on a moving charge in a magnetic field, F = q·v × B, is perpendicular to both the velocity and the field, and why a motor turns one way and not the other.

The scalar triple product u · (v × w) is a number: the determinant of the matrix of the three vectors’ components, and in absolute value, the volume of the parallelepiped. If it is zero, the three vectors are coplanar, which is the same as saying they are linearly dependent.

06Seeing it in the plane

Drag the tips of the two vectors. Everything above is recalculated live: the sum by the parallelogram, the dot product, the angle, the projection and the area they define.

Lab · two vectors in the plane

The vectors

Products

07Vectors in electronics

Phasors. A sinusoidal voltage is represented by a rotating vector; adding two signals of the same frequency means adding vectors. This is the subject of the vector study of alternating current.
Power. The power triangle —active, reactive and apparent— is a vector sum: S² = P² + Q². The active power comes from the dot product between voltage and current; the reactive power, from the perpendicular part.
Fields. The electric and magnetic fields are vector fields, and Maxwell’s equations are written with dot and cross products. The Poynting vector, which indicates where the energy of a wave flows, is S = E × H.
Mechanics in the lab. Forces on a bracket, moments in a motor, accelerations: all of Physics I is applied vector algebra.

08In the lab

Exercise 1 · Angle between two vectors

Given u = (2, 3, −1) and v = (−1, 4, 2), compute by hand the dot product, the magnitudes, the angle and the projection of u onto v. Verify the angle with the cross product: |u × v| = |u||v| sin θ has to give the same θ.

Exercise 2 · Area and volume

Compute the area of the triangle with vertices A(1,0,2), B(3,1,0) and C(0,2,1) as half the magnitude of the cross product of two of its sides. Then add a fourth point and use the scalar triple product to decide whether the four are coplanar.

Exercise 3 · The right-hand rule with the multimeter

With a coil and a magnet, check the direction of the induced voltage when moving the magnet and compare it with what the cross product in Faraday’s law predicts. An oscilloscope or the instrument simulator helps you see the sign of the pulse.

09Common mistakes

Adding magnitudes. |u + v| is not |u| + |v| unless they are parallel and have the same sense.
Confusing the two products: the dot product gives a number, the cross product a vector.
Reversing the order in the cross product and ending up with the sense flipped.
Using the cross product in the plane without stating that you are working in R³ with z = 0.
Forgetting to normalize when projecting: the projection has |v|² in the denominator, not |v|.
Mixing degrees and radians when computing the angle with the calculator.

10Self-assessment

Compute u · v for u = (2, 3, −1) and v = (−1, 4, 2).

−2 + 12 − 2 = 8. Since it is positive, the angle between them is acute.

What is the unit vector of (0, 3, 4)?

Its magnitude is 5, so the unit vector is (0, 0.6, 0.8).

When is the cross product zero?

When the vectors are parallel or antiparallel, or when either one is the zero vector. Then sin θ = 0 and the parallelogram degenerates.

Compute î × ĵ and ĵ × î.

î × ĵ = k̂ and ĵ × î = −k̂: the cross product is anticommutative.

If u · v = 0 and neither is zero, what angle do they form?

90°: they are perpendicular.

What does it mean for the scalar triple product of three vectors to be zero?

That they are coplanar: the parallelepiped has zero volume and the three are linearly dependent.

11Further reading

Charles H. Lehmann. Analytic Geometry. (Spanish edition, “Geometría analítica”), Limusa. The classic for the geometric part: vectors, lines, planes and conics in great detail.
Stanley I. Grossman. Álgebra lineal. 6th ed., McGraw-Hill, 2008 (in Spanish). Treats the vectors of R² and R³ as a gateway to vector spaces.
Murray R. Spiegel. Schaum’s Outline of Vector Analysis. 2nd ed. (Spanish edition, “Análisis vectorial”, Schaum series), McGraw-Hill, 2009. For practice: hundreds of worked exercises, and it goes on to gradient, divergence and curl.
Raymond A. Serway and John W. Jewett. Physics for Scientists and Engineers. 9th ed. (Spanish edition, “Física para ciencias e ingeniería”), Cengage Learning, 2014. Where they are really used: forces, fields, torques and the cross product in magnetism.

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