Scalars and Vectors in Mechanics

Mechanics uses two different kinds of physical quantities repeatedly. Some quantities are completely specified by a magnitude, while others require both magnitude and direction. The distinction is not merely graphical notation: it determines how physical quantities combine, how their components change when axes are changed, and which mathematical operations are physically meaningful.

A scalar is a physical quantity represented by a single value that is independent of the orientation of the coordinate axes. A vector is a geometric quantity with magnitude and direction whose components transform together when the coordinate basis is changed.

Examples of scalars in mechanics include mass, time, temperature, distance, speed, energy, and power. Examples of vectors include position, displacement, velocity, acceleration, momentum, force, torque, and angular momentum.

The main ideas of this article are:

1. distinguish scalars from vectors and vectors from their components;
2. represent vectors using magnitude, direction, and Cartesian components;
3. add and subtract vectors geometrically and componentwise;
4. use unit vectors to encode coordinate directions;
5. understand the dot product as a scalar measure of directional overlap;
6. understand the cross product as a vector measure of perpendicular action;
7. connect these operations to work, projection, torque, and angular momentum.

1 Scalars

A scalar quantity is described by one numerical value together with its unit. If a particle has mass

(1)

then no spatial direction is needed to specify the mass.

Likewise, a kinetic energy

(2)

is a scalar. Energy can increase or decrease, but it does not point north, upward, or along a coordinate axis.

A scalar may be positive, negative, or zero when the physical definition permits signed values. For example, an electric potential or a one-dimensional coordinate may be signed. Being scalar does not mean being positive; it means that a spatial orientation is not part of the quantity itself.

2 Vectors

A vector has both magnitude and direction. A displacement from one location to another is a vector because both the distance and the direction of the change matter. A velocity is a vector because two objects moving at the same speed in different directions do not have the same velocity.

A vector is written here in boldface, for example

(3)

Its magnitude is written

(4)

The magnitude is a scalar and is nonnegative.

2.1 A vector is not the same thing as its components

Suppose a vector is represented in a Cartesian basis by

(5)

The numbers A_x, A_y, and A_z are components of the vector relative to the chosen basis. If the coordinate axes are rotated, these components generally change even though the physical vector does not.

This distinction is important in mechanics: a force does not physically change merely because the analyst chooses a different set of axes.

3 Unit vectors and Cartesian components

The Cartesian unit vectors

(6)

all have magnitude one and point along the positive coordinate directions.

For an orthonormal Cartesian basis,

(7)

(8)

(9)

and different basis vectors are perpendicular, so

(10)

with analogous relations for the other pairs.

A two-dimensional vector may be written

(11)

Its magnitude is

(12)

In three dimensions,

(13)

4 Resolving a vector into components

Suppose a vector A lies in the xy plane and makes an angle 𝜃 measured counterclockwise from the positive x axis. Then

(14)

(15)

Hence

(16)

If the components are known, the direction can be reconstructed using trigonometry. The safest computational form is a quadrant-aware two-argument arctangent. Conceptually,

(17)

but the signs of both components must be used to place 𝜃 in the correct quadrant.

5 Worked example 1: force components and reconstruction

A cable exerts a force of magnitude

(18)

at an angle of 35^∘ above the positive horizontal direction. Find its Cartesian components. Then verify that the components reproduce the original magnitude.

The horizontal component is

(19)

Numerically,

(20)

The vertical component is

(21)

so

(22)

Thus

(23)

Reconstructing the magnitude,

(24)

which gives approximately

(25)

The component representation and the magnitude-angle representation describe the same vector.

6 Vector addition and subtraction

Vectors add geometrically by the head-to-tail rule. If an object undergoes displacement A followed by displacement B, the net displacement is

(26)

Componentwise,

(27)

(28)

and similarly in three dimensions.

Vector subtraction is defined by adding the negative vector:

(29)

The vector −B has the same magnitude as B and points in the opposite direction.

Vector addition is commutative,

(30)

and associative,

(31)

7 Worked example 2: two displacement vectors

A hiker walks 3.0 km east and then 4.0 km north. Find the net displacement magnitude and direction.

Choose east as +x and north as +y. The two displacements are

(32)

(33)

Therefore

(34)

The magnitude is

(35)

The direction satisfies

(36)

so

(37)

The hiker is therefore displaced 5.0 km at 53.1^∘ north of east.

This example illustrates why traveled distance and displacement are different. The total distance walked is 7.0 km, a scalar, while the displacement is a 5.0 km vector.

8 Multiplication of a vector by a scalar

If c is a scalar, then

(38)

means every component of A is multiplied by c. If c > 0, the direction is unchanged. If c < 0, the direction reverses.

For example, momentum for a particle of mass m is

(39)

The scalar mass multiplies the velocity vector, so momentum points in the same direction as velocity for positive mass.

9 The dot product

The dot product of two vectors is a scalar:

(40)

where 𝜃 is the angle between the vectors.

In Cartesian components,

(41)

Several important special cases follow immediately. If two nonzero vectors are perpendicular,

(42)

so

(43)

If they point in the same direction,

(44)

If they point in opposite directions,

(45)

9.1 Projection

The scalar component of A along a unit vector e is

(46)

The corresponding vector projection is

(47)

This is the mathematical structure behind resolving gravity along an incline, finding the component of velocity along a line, and determining the component of force along a displacement.

9.2 Work as a dot product

For a constant force acting through displacement Δr, the work is

(48)

Thus

(49)

Only the component of force parallel to the displacement contributes to work.

10 Worked example 3: work from a force at an angle

A person pulls a crate through a horizontal displacement of 5.0 m using a constant force of magnitude 40 N directed 30^∘ above the horizontal. Find the work done by this pulling force.

The displacement is horizontal, so the angle between force and displacement is 30^∘. Therefore

(50)

Substituting,

(51)

so

(52)

The vertical component of the pulling force contributes no work in this idealized example because the crate has no vertical displacement.

11 The cross product

The cross product of two vectors is a vector perpendicular to the plane containing them. Its magnitude is

(53)

Its direction is set by the right-hand rule.

In a right-handed Cartesian basis,

(54)

(55)

(56)

Reversing the order changes the sign:

(57)

A vector crossed with itself is zero.

For

(58)

and

(59)

the component form is

(60)

11.1 Torque and angular momentum

Torque about an origin is

(61)

Its magnitude is

(62)

Equivalently,

(63)

where d_⊥ is the perpendicular lever arm from the origin to the line of action of the force.

Angular momentum of a particle is

(64)

Both quantities depend on cross products because the rotational effect is associated with perpendicular geometry. More precisely, torque and angular momentum are axial vectors (sometimes called pseudovectors): under ordinary rotations they behave like vectors, while reflections introduce an additional sign distinction. That refinement is deferred to later three-dimensional mechanics.

12 Worked example 4: torque from a force on a wrench

A force of magnitude 80 N is applied to a wrench at a point 0.25 m from the bolt. The angle between the position vector from the bolt to the hand and the applied force is 60^∘. Find the torque magnitude.

Using

(65)

we obtain

(66)

Therefore

(67)

If the force were applied parallel to the wrench handle, then 𝜃 = 0 and the torque would vanish even though the force itself is nonzero. Rotational effectiveness depends on the perpendicular component.

13 Angle between two vectors

If A and B are nonzero, the dot product can be used to determine the angle between them:

(68)

Therefore

(69)

This relation is useful when the vectors are given by components rather than by a diagram.

14 Why vector equations are powerful

A vector equation packages several component equations into one geometric statement. Newton’s second law,

(70)

means simultaneously that

(71)

(72)

and

(73)

The vector equation is independent of a particular Cartesian orientation. The component equations depend on the chosen axes, but the physical relation does not.

15 Scalars formed from vectors

Many important scalars are constructed from vectors. Examples include speed,

(74)

kinetic energy,

(75)

and work,

(76)

The dot product is especially important because it produces a scalar that is independent of a rotation of Cartesian axes.

16 Common mistakes

1. Confusing speed and velocity. Speed is the scalar magnitude of the velocity vector.
2. Confusing distance and displacement. Distance is total path length; displacement is the vector from initial to final position.
3. Adding vector magnitudes instead of components. Magnitudes add directly only when vectors are collinear and oriented consistently.
4. Using sine and cosine without defining the angle. Components depend on which axis the angle is measured from.
5. Ignoring quadrants when reconstructing direction. The signs of both components matter.
6. Treating components as physical vectors. Components depend on the chosen coordinate basis.
7. Assuming the dot product is a vector. It is a scalar.
8. Assuming the cross product is commutative. Reversing the order reverses the result.
9. Forgetting units. Vector operations do not remove physical dimensions; for example, torque has units N m.
10. Using a cross product for parallel projection. Projection is naturally a dot-product operation.

17 Problem-solving workflow

For a vector mechanics problem, a reliable workflow is:

1. draw the geometry and define the coordinate axes;
2. identify which quantities are scalars and which are vectors;
3. resolve vectors into components using clearly defined angles;
4. perform vector addition or subtraction componentwise;
5. use the dot product for parallel projection, work, and angle tests;
6. use the cross product for torque, angular momentum, and perpendicular orientation;
7. reconstruct a magnitude or direction only after the component calculation is complete;
8. inspect units, signs, quadrants, and limiting cases.

18 Practice problems

The following problems use stable mechanics identifiers. Short answers are supplied after the GRE-speed questions.

MECH-VEC-001 — Scalar or vector?

Classify each quantity as scalar or vector: mass, displacement, distance, speed, velocity, acceleration, kinetic energy, momentum, force, work, torque, and power.

MECH-VEC-002 — Components from magnitude and direction

A displacement has magnitude 12 m and points 40^∘ above the positive x axis. Find its x and y components.

MECH-VEC-003 — Magnitude and direction from components

A velocity is

(77)

Find its speed and direction measured counterclockwise from the positive x axis.

MECH-VEC-004 — Resultant displacement

A robot moves 5 m east, 2 m north, and then 3 m west. Find the resultant displacement vector and its magnitude.

MECH-VEC-005 — Vector addition in three dimensions

Let

(78)

and

(79)

Find A + B and A − B.

MECH-VEC-006 — Dot product and angle

For

(80)

and

(81)

compute A ⋅ B and determine the angle between the vectors.

MECH-VEC-007 — Projection along an incline

A gravitational force has magnitude mg vertically downward. An incline is at angle 𝜃 above the horizontal. Show that the component of gravity down the incline has magnitude mg sin 𝜃.

MECH-VEC-008 — Work by a constant force

A force

(82)

acts through displacement

(83)

Find the work done.

MECH-VEC-009 — Cross product

Let

(84)

and

(85)

Find A × B.

MECH-VEC-010 — Torque in components

A force

(86)

is applied at position

(87)

Find the torque about the origin.

MECH-VEC-011 — Momentum and impulse direction

A 2 kg particle has velocity

(88)

Find its momentum vector and magnitude. If an impulse 8e_y N s is applied, find the final momentum.

MECH-VEC-012 — Unit vector along a direction

Find a unit vector in the direction of

(89)

Verify that your result has magnitude one.

19 GRE-speed questions

MECH-GRE-VEC-001

Which quantity is a vector?

1. kinetic energy
2. speed
3. momentum
4. work

MECH-GRE-VEC-002

A vector of magnitude 10 points 60^∘ above the positive x axis. Its x component is

1. 5
2. 5
3. 10
4. 10

MECH-GRE-VEC-003

If two nonzero vectors satisfy

(90)

then the vectors are

1. parallel
2. antiparallel
3. perpendicular
4. equal in magnitude

MECH-GRE-VEC-004

The magnitude of A × B is largest when the angle between A and B is

1. 0^∘
2. 30^∘
3. 60^∘
4. 90^∘

MECH-GRE-VEC-005

A force is perpendicular to a displacement. The work done by that force is

1. positive
2. negative
3. zero
4. equal to FΔr

MECH-GRE-VEC-006

A force acts directly along the position vector from a pivot to its point of application. The torque about the pivot is

1. maximum
2. zero
3. equal to rF
4. impossible to determine

20 Short answer key

Practice problems:

1. Scalars: mass, distance, speed, kinetic energy, work, power. Vectors: displacement, velocity, acceleration, momentum, force, torque.
2. A_x = 9.19 m, A_y = 7.71 m.
3. Speed 10 m/s; direction 126.9^∘ counterclockwise from +x.
4. R = 2e_x + 2e_y m; R = 2 = 2.83 m.
5. A + B = −2e_x + 4e_y + 4e_z; A − B = 6e_x − 6e_y + 2e_z.
6. Dot product 0; angle 90^∘.
7. mg sin 𝜃 down the incline.
8. W = 38 J.
9. A × B = −14e_z.
10. τ = 6e_z N m.
11. p_i = 6e_x − 8e_y kg m/s, |p_i| = 10 kg m/s; p_f = 6e_x kg m/s.
12. e_A = (2e_x − 2e_y + e_z)∕3.

GRE-speed answers:

1. C
2. A
3. C
4. D
5. C
6. B

21 Connections to later mechanics

This article provides the vector language used throughout the mechanics curriculum. The next foundations article develops coordinate systems. Later articles apply these tools to kinematics, Free-body diagrams, Newton’s laws, work and energy, momentum, angular momentum, central-force motion, rotating frames, and rigid-body mechanics.

References