Unit 1 · Topic 02

Vectors

Vectors represent quantities whose description requires magnitude and direction. Their components depend on the chosen basis; the physical vector does not.

Unit 1VectorsOpen navigation

Concept 01

Scalars and vectors

Essential The minimum you should retain

A vector is a quantity with magnitude and direction. Its sense distinguishes the two possible orientations along the same direction.

A scalar is described by a value and its unit. Temperature, time, and mass are scalars; displacement and velocity are vectors.

UnderstandInterpret and connect

Two vectors are equal when they have the same magnitude, direction, and sense, even if they are drawn at different locations.

A vector's magnitude is never negative. A component may be negative because it compares the vector with the positive sense of an axis.

DeepenFormulation and conditions

A geometric vector can be represented in different bases. Components change with the basis, while magnitudes and geometric relations remain invariant.

ExploreConnections for further study

This independence from the basis allows the same displacement to be described with different axes. Choosing a convenient basis simplifies the calculation but does not change the physical vector.

Concept 02

Cartesian components, unit vectors, and bases

Essential The minimum you should retain

Cartesian components indicate how much of a vector points along each axis. Their signs depend on the chosen positive directions.

The unit vectors i, j, and k provide reference directions for the x, y, and z axes.

UnderstandInterpret and connect

In a three-dimensional Cartesian basis, A→=Axi^+Ayj^+Azk^ \vec A=A_x\hat i+A_y\hat j+A_z\hat k . If θ is measured from +x +x in the plane, Ax A_{x} = A cos θ and Ay A_{y} = A sin θ; geometry and quadrant determine the signs.

The function atan2(Ay,Ax) \operatorname{atan2}(A_y,A_x) preserves quadrant information; using only arctan(AyAx) \arctan(A_y/A_x) can produce an incorrect direction.

DeepenFormulation and conditions

Magnitude follows from the extended Pythagorean theorem. Recovering direction requires combining the components with their signs.

ExploreConnections for further study

Components are projections onto a basis. Reconstruction is still possible in a nonorthogonal basis, but the Pythagorean theorem can no longer be applied directly to the components.

Mathematical relation

Magnitude of a three-dimensional Cartesian vector

A=Ax2+Ay2+Az2 A=\sqrt{A_x^2+A_y^2+A_z^2}
Represents

The length of vector A in an orthonormal Cartesian basis.

Physical interpretation

The magnitude is nonnegative even when a component is negative.

DeepenVariables, conditions, and checks

Variables

Ax A_{x} , Ay A_{y} , Az A_{z}
Cartesian components of A; usual unit: same unit as A

Conditions of application

  • The basis axes are mutually perpendicular and the unit vectors have magnitude one.

Dimensional check

All squared terms must have the same dimension.

Errors it helps prevent

  • Interpreting a negative component as a negative magnitude.

Mathematical relation

Components of a plane vector from its magnitude and angle

Ax=Acosθ,Ay=Asinθ A_x=A\cos\theta,\quad A_y=A\sin\theta
Represents

The Cartesian projections of A in the plane.

Physical interpretation

Cosine accompanies the leg adjacent to the defined angle; sine accompanies the opposite leg.

DeepenVariables, conditions, and checks

Variables

A
magnitude of the vector; usual unit: unit of A
θ
angle measured from +x +x ; usual unit: rad or °

Conditions of application

  • θ is measured from the +x +x axis and the signs are determined by the quadrant.

Dimensional check

Ax A_{x} and Ay A_{y} have the same dimension as A; sine and cosine are dimensionless.

Errors it helps prevent

  • Swapping sine and cosine without checking the axis from which θ is measured.
Vector A starts at the origin and ends in the first quadrant. Component A x is horizontal and component A y completes the vector vertically.AAxAyO

Vector A does not change when it is decomposed: Ax A_{x} and Ay A_{y} are signed projections on the chosen axes and form the same resultant displacement.

Concept 03

Vector addition and subtraction

Essential The minimum you should retain

To add vectors, add corresponding components. Geometrically, place the second vector at the tip of the first; the resultant runs from the initial origin to the final tip.

Subtracting B is equivalent to adding the opposite vector −B→ -\vec B .

UnderstandInterpret and connect

Vector addition is commutative and associative. Construction order may change the intermediate drawing, but not the resultant.

A small resultant does not mean the vectors are small: they may have large magnitudes and nearly opposite senses.

DeepenFormulation and conditions

The component method works in any dimension and does not depend on drawing scale. A diagram remains useful for anticipating signs and approximate direction.

ExploreConnections for further study

Adding many vectors can be represented by a head-to-tail polygon. If the polygon closes, the resultant is zero—an idea that will later be useful when studying equilibrium.

A starts at the origin. B starts at the head of A. The resultant A plus B connects the origin to the final head.ABA + BO

The head-to-tail rule preserves each vector. The resultant A+B connects the first vector's origin to the second vector's head.

Concept 04

Dot product and cross product

Essential The minimum you should retain

The dot product A→·B→ \vec A\cdot\vec B is a number that measures how much one vector points along the other. It is zero when nonzero vectors are perpendicular.

The cross product A→×B→ \vec A\times\vec B produces a vector perpendicular to the plane of A and B. Its sense is determined by the right-hand rule.

UnderstandInterpret and connect

The magnitude AB cos θ of the dot product can be viewed as one vector's magnitude times the projection of the other along it.

The magnitude AB sin θ of the cross product equals the area of the parallelogram formed by the vectors.

DeepenFormulation and conditions

The dot product is commutative. The cross product is not: B→×A→=−(A→×B→) \vec B\times\vec A=-(\vec A\times\vec B) . Parallel vectors have zero cross product.

ExploreConnections for further study

These products connect geometry with other physical quantities: the dot product appears in work, while the cross product appears in torque and angular momentum.

Mathematical relation

Dot product of two vectors

A→·B→=ABcosθ \vec A\cdot\vec B=AB\cos\theta
Represents

A scalar measure of the alignment between A and B.

Physical interpretation

It is positive for an acute angle, zero for perpendicular vectors, and negative for an obtuse angle.

DeepenVariables, conditions, and checks

Variables

θ
smaller angle between A and B; usual unit: rad or °

Conditions of application

  • A and B are measured in Euclidean space with the same unit system for each component.

Dimensional check

It has the dimension of the product of the magnitudes A and B.

Errors it helps prevent

  • Forgetting that the result is a scalar.

Mathematical relation

Magnitude of the cross product of two vectors

|A→×B→|=ABsinθ |\vec A\times\vec B|=AB\sin\theta
Represents

The magnitude of the vector perpendicular to A and B.

Physical interpretation

The magnitude equals the area of the parallelogram built from A and B.

DeepenVariables, conditions, and checks

Variables

θ
smaller angle between A and B; usual unit: rad or °

Conditions of application

  • The direction is determined with the right-hand rule.

Dimensional check

It has the dimension of the product of the magnitudes A and B.

Errors it helps prevent

  • Treating A→×B→ \vec A\times\vec B as commutative; B→×A→ \vec B\times\vec A points in the opposite direction.
Vector B lies on the horizontal axis. Vector A forms an acute angle, and a perpendicular guide shows its projection onto B.ABprojBA

The projection of A onto the direction of B measures the part of A aligned with B. The dot product multiplies this projection by the magnitude of B.

Concept review

Common errors

Each warning includes a concrete way to review the reasoning, not only an incorrect-answer marker.

Confusing a vector's magnitude with one of its components.

A component is a signed projection; magnitude combines all components and is nonnegative.

Interpreting a negative component as a negative magnitude.

A component's sign indicates orientation relative to an axis. The vector's magnitude remains nonnegative.

Swapping sine and cosine without analysing the geometry.

Identify the axis from which the angle is measured and the adjacent leg before writing components.

Choosing the wrong quadrant when recovering a direction.

Use the signs of both components and a quadrant-preserving function such as atan2.

Treating the cross product as commutative.

Reversing the order reverses the direction: B→×A→=−(A→×B→) \vec B\times\vec A=-(\vec A\times\vec B) .