A vector has magnitude and direction. The zero vector \(\mathbf0\) has magnitude zero and no defined direction; it is treated as parallel to every vector only by the algebraic scalar-multiple convention. IdeaVector statement Equality\(\mathbf a=\mathbf b\) when corresponding components are equal. Equal vectors have the same magnitude and direction. Negative\(-\mathbf a\) has the same magnitude as \(\mathbf a\) and the opposite direction. AdditionPlace vectors head-to-tail, or use the parallelogram law. Subtraction\(\mathbf a-\mathbf b=\mathbf a+(-\mathbf b)\). Scalar multiple\(k\mathbf a\) has magnitude \(|k||\mathbf a|\); its direction reverses when \(k<0\). Triangle law Parallelogram law Vector subtraction Scalar multiplication Resolve a position vector along the mutually perpendicular unit directions \(\mathbf i,\mathbf j,\mathbf k\). Component calculations reduce vector algebra to ordinary scalar algebra. Component forms \[\mathbf r=x\mathbf i+y\mathbf j=\begin{pmatrix}x\\y\end{pmatrix},\qquad \mathbf r=x\mathbf i+y\mathbf j+z\mathbf k=\begin{pmatrix}x\\y\\z\end{pmatrix}\] LawStatement Commutative addition\(\mathbf a+\mathbf b=\mathbf b+\mathbf a\) Associative addition\((\mathbf a+\mathbf b)+\mathbf c=\mathbf a+(\mathbf b+\mathbf c)\) Distributive\(k(\mathbf a+\mathbf b)=k\mathbf a+k\mathbf b\) Distributive in scalars\((k+m)\mathbf a=k\mathbf a+m\mathbf a\) Associative scalar multiplication\(k(m\mathbf a)=(km)\mathbf a\) Two-dimensional components Three-dimensional components \(\mathbf r=x\mathbf i+y\mathbf j+z\mathbf k\). Axis directions have two zero components: the \(x\)-, \(y\)- and \(z\)-axes use directions \(\mathbf i,\mathbf j,\mathbf k\). A vector parallel to the \(xy\)-, \(xz\)- or \(yz\)-plane has respectively zero \(k\)-, \(j\)- or \(i\)-component. Typed solution erratum Correction to the accompanying typed transcription: the original handwritten solution shows \(\mathbf b=\begin{pmatrix}2\\1\\-1\end{pmatrix}\), with magnitude \(\sqrt{2^2+1^2+(-1)^2}=\sqrt6\). The typed column vector omitted the middle component 1; the original magnitude calculation is correct. The original route for BD is \(\overrightarrow{BD}=\overrightarrow{BO}+\overrightarrow{OD}=-\overrightarrow{OB}+\overrightarrow{OD}=\mathbf d-\mathbf b\); the typed explanation inserted an extra \(\overrightarrow{OB}\). The original handwritten solution, recorded example and media are retained.

Vector Fundamentals and Components

A vector has magnitude and direction. The zero vector \(\mathbf0\) has magnitude zero and no defined direction; it is treated as parallel to every vector only by the algebraic scalar-multiple convention.

Idea	Vector statement
Equality	\(\mathbf a=\mathbf b\) when corresponding components are equal. Equal vectors have the same magnitude and direction.
Negative	\(-\mathbf a\) has the same magnitude as \(\mathbf a\) and the opposite direction.
Addition	Place vectors head-to-tail, or use the parallelogram law.
Subtraction	\(\mathbf a-\mathbf b=\mathbf a+(-\mathbf b)\).
Scalar multiple	\(k\mathbf a\) has magnitude \(\|k\|\|\mathbf a\|\); its direction reverses when \(k<0\).

Triangle law

Parallelogram law

Vector subtraction

Scalar multiplication

Resolve a position vector along the mutually perpendicular unit directions \(\mathbf i,\mathbf j,\mathbf k\). Component calculations reduce vector algebra to ordinary scalar algebra.

Component forms

\[\mathbf r=x\mathbf i+y\mathbf j=\begin{pmatrix}x\\y\end{pmatrix},\qquad \mathbf r=x\mathbf i+y\mathbf j+z\mathbf k=\begin{pmatrix}x\\y\\z\end{pmatrix}\]

Law	Statement
Commutative addition	\(\mathbf a+\mathbf b=\mathbf b+\mathbf a\)
Associative addition	\((\mathbf a+\mathbf b)+\mathbf c=\mathbf a+(\mathbf b+\mathbf c)\)
Distributive	\(k(\mathbf a+\mathbf b)=k\mathbf a+k\mathbf b\)
Distributive in scalars	\((k+m)\mathbf a=k\mathbf a+m\mathbf a\)
Associative scalar multiplication	\(k(m\mathbf a)=(km)\mathbf a\)

Two-dimensional components

Three-dimensional components

\(\mathbf r=x\mathbf i+y\mathbf j+z\mathbf k\).

Axis directions have two zero components: the \(x\)-, \(y\)- and \(z\)-axes use directions \(\mathbf i,\mathbf j,\mathbf k\). A vector parallel to the \(xy\)-, \(xz\)- or \(yz\)-plane has respectively zero \(k\)-, \(j\)- or \(i\)-component.

Typed solution erratum

Correction to the accompanying typed transcription: the original handwritten solution shows \(\mathbf b=\begin{pmatrix}2\\1\\-1\end{pmatrix}\), with magnitude \(\sqrt{2^2+1^2+(-1)^2}=\sqrt6\). The typed column vector omitted the middle component 1; the original magnitude calculation is correct. The original route for BD is \(\overrightarrow{BD}=\overrightarrow{BO}+\overrightarrow{OD}=-\overrightarrow{OB}+\overrightarrow{OD}=\mathbf d-\mathbf b\); the typed explanation inserted an extra \(\overrightarrow{OB}\). The original handwritten solution, recorded example and media are retained.

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