Vector Algebra — Basic Definitions

I.1 Components of a Vector

If a vector $\overline{a}$ has Cartesian rectangular coordinates $x,\,y,\,z$ , then

\overline{a}=x\overline{i}+y\overline{j}+z\overline{k}=\{x,\,y,\,z\}.

If a vector $\overline{AB}$ is given by its initial point $A(x_1,y_1,z_1)$ and terminal point $B(x_2,y_2,z_2)$ , then the coordinates of this vector $\overline{AB}=\{x,y,z\}$ equal the differences of the like coordinates of the end and the start:

x=x_2-x_1;\qquad y=y_2-y_1;\qquad z=z_2-z_1.

The zero vector is a vector whose start and end coincide. It has magnitude zero and no defined direction.

I.2 Sum and Difference of Vectors

Sum. The sum of two vectors is built either by the triangle rule (place them head to tail) or by the parallelogram rule (draw them from a common origin).

Triangle rule: vector a followed by vector b, resultant c = a + b

Parallelogram rule: vectors a and b from a common origin, diagonal c = a + b

Difference. The difference $\overline{c}=\overline{a}-\overline{b}$ is the other diagonal of the parallelogram built on $\overline{a}$ and $\overline{b}$ .

Difference of vectors: parallelogram on a and b, with c = a − b

If $\overline{a}=\{x_1,y_1,z_1\}$ and $\overline{b}=\{x_2,y_2,z_2\}$ , then

\overline{a}\pm\overline{b}=\{x_1\pm x_2;\ y_1\pm y_2;\ z_1\pm z_2\}.

If $\overline{a}=\{x_1,y_1,z_1\}$ and $\lambda=\text{const}$ , then

\lambda\overline{a}=\{\lambda x_1,\ \lambda y_1,\ \lambda z_1\}.

I.3 Magnitude of a Vector

If $\overline{a}=\{x,y,z\}$ , the magnitude (length) of the vector in an orthonormal basis is

|\overline{a}|=\sqrt{x^2+y^2+z^2}.

Since

\begin{aligned}\cos\alpha&=\cos\bigl(\widehat{\overline{a},\overline{i}}\bigr)=\frac{x}{|\overline{a}|};\\[6pt] \cos\beta&=\cos\bigl(\widehat{\overline{a},\overline{j}}\bigr)=\frac{y}{|\overline{a}|};\\[6pt] \cos\gamma&=\cos\bigl(\widehat{\overline{a},\overline{k}}\bigr)=\frac{z}{|\overline{a}|},\end{aligned}

we have $\cos^2\alpha+\cos^2\beta+\cos^2\gamma=1$ . The direction cosines of a nonzero vector are the coordinates of its unit vector:

\overline{a}^{\,0}=\frac{\overline{a}}{|\overline{a}|}=(\cos\alpha,\ \cos\beta,\ \cos\gamma)\qquad(\overline{a}\neq\overline{0}).

The zero vector has no direction, hence no direction cosines and no unit vector.

I.4 Scalar Product of Vectors

Definition. The scalar product (dot product) of two vectors is the number equal to the product of their lengths and the cosine of the angle between them:

(\overline{a}\cdot\overline{b})=|\overline{a}|\cdot|\overline{b}|\cos\bigl(\widehat{\overline{a}\,\overline{b}}\bigr).

Properties of the scalar product:

$\overline{a}\cdot\overline{b}=\overline{b}\cdot\overline{a}$ — commutativity;
$(\lambda\overline{a})\cdot\overline{b}=\overline{a}\cdot(\lambda\overline{b})=\lambda(\overline{a}\cdot\overline{b})$ — homogeneity (a scalar factor pulls out); the dot product is not associative, since $(\overline{a}\cdot\overline{b})\cdot\overline{c}$ is undefined;
$\overline{a}\cdot(\overline{b}+\overline{c})=\overline{a}\cdot\overline{b}+\overline{a}\cdot\overline{c}$ — distributivity;
$\overline{a}\cdot\overline{b}=0\Leftrightarrow\overline{a}\perp\overline{b}$ , or $\overline{a}=0$ , or $\overline{b}=0$ ;
$\overline{a}\cdot\overline{a}=|\overline{a}|^2\Rightarrow|\overline{a}|=\sqrt{\overline{a}\cdot\overline{a}}$ .

If $\overline{a}=\{x_1,y_1,z_1\}$ and $\overline{b}=\{x_2,y_2,z_2\}$ , then

(\overline{a}\cdot\overline{b})=x_1x_2+y_1y_2+z_1z_2.

I.4 (a) Length of

\overline{a}=\alpha\overline{m}+\beta\overline{n}

If $\overline{a}=\alpha\overline{m}+\beta\overline{n}$ , where $|\overline{m}|,\,|\overline{n}|$ and the angle $\widehat{\overline{m},\overline{n}}$ between the vectors are known and $\alpha,\beta=\text{const}$ , then

\begin{aligned}|\overline{a}|&=\sqrt{\overline{a}\cdot\overline{a}}=\sqrt{(\alpha\overline{m}+\beta\overline{n})\cdot(\alpha\overline{m}+\beta\overline{n})}=\\[6pt] &=\sqrt{\alpha^2(\overline{m}\cdot\overline{m})+2\alpha\beta(\overline{m}\cdot\overline{n})+\beta^2(\overline{n}\cdot\overline{n})}=\\[6pt] &=\sqrt{\alpha^2|\overline{m}|^2+2\alpha\beta|\overline{m}||\overline{n}|\cos\bigl(\widehat{\overline{m},\overline{n}}\bigr)+\beta^2|\overline{n}|^2}.\end{aligned}

I.4 (b) Scalar Product of Two Combinations

If $\overline{a}=\alpha\overline{m}+\beta\overline{n}$ and $\overline{b}=l\overline{m}+d\overline{n}$ , where $|\overline{m}|,\,|\overline{n}|$ and the angle $\widehat{\overline{m},\overline{n}}$ are known and $\alpha,\beta,l,d=\text{const}$ , then

\begin{aligned}(\overline{a}\cdot\overline{b})&=(\alpha\overline{m}+\beta\overline{n})\cdot(l\overline{m}+d\overline{n})=\alpha l\,|\overline{m}|^2+\\[6pt] &\quad+(\alpha d+\beta l)\,|\overline{m}||\overline{n}|\cos\bigl(\widehat{\overline{m},\overline{n}}\bigr)+\beta d\,|\overline{n}|^2.\end{aligned}

I.4 (c) Angle Between Two Vectors

The angle $\varphi$ between two nonzero vectors $\overline{a}$ and $\overline{b}$ is computed by the formula

\cos\varphi=\frac{(\overline{a}\cdot\overline{b})}{|\overline{a}|\cdot|\overline{b}|}\qquad(\overline{a},\overline{b}\neq\overline{0}).

Two vectors a and b with the angle φ between them

I.4 (d) Angle Between Two Vectors in Coordinates

If $\overline{a}=\{x_1,y_1,z_1\}$ and $\overline{b}=\{x_2,y_2,z_2\}$ are both nonzero, then

\cos\varphi=\frac{(\overline{a}\cdot\overline{b})}{|\overline{a}||\overline{b}|}=\frac{x_1x_2+y_1y_2+z_1z_2}{\sqrt{x_1^2+y_1^2+z_1^2}\,\sqrt{x_2^2+y_2^2+z_2^2}}.

I.5 Vector Product of Vectors

Definition. The vector product (cross product) of a vector $\overline{a}$ by a vector $\overline{b}$ is the vector $\overline{c}$ satisfying the following conditions:

$|\overline{c}|=|\overline{a}|\cdot|\overline{b}|\sin\bigl(\widehat{\overline{a},\overline{b}}\bigr)$ ;
$\overline{c}\perp\overline{a},\ \overline{c}\perp\overline{b}$ ;
the vector $\overline{c}$ is directed so that, seen from its end, the shortest rotation from $\overline{a}$ to $\overline{b}$ is counterclockwise (the right-hand rule).

The vector product is denoted $\overline{a}\times\overline{b}$ or $[\overline{a},\overline{b}]$ . Conditions 2–3 fix a direction only when $\overline{a}\times\overline{b}\neq\overline{0}$ , i.e. when $\overline{a}$ and $\overline{b}$ are nonzero and non-parallel; otherwise $\overline{a}\times\overline{b}=\overline{0}$ (property 2 below).

Properties of the vector product:

$\overline{a}\times\overline{b}=-\,\overline{b}\times\overline{a}$ — anticommutative;
$\overline{a}\times\overline{b}=0$ if $\overline{a}=0$ or $\overline{b}=0$ , or $\overline{a}\parallel\overline{b}$ ;
$(\lambda\overline{a})\times\overline{b}=\overline{a}\times(\lambda\overline{b})=\lambda(\overline{a}\times\overline{b})$ — the scalar factor may be taken outside;
$\overline{a}\times(\overline{b}+\overline{c})=\overline{a}\times\overline{b}+\overline{a}\times\overline{c}$ — distributive over addition.

If $\overline{a}=\{x_1,y_1,z_1\}$ and $\overline{b}=\{x_2,y_2,z_2\}$ , then

\begin{aligned}\overline{a}\times\overline{b}&=\begin{vmatrix}\overline{i}&\overline{j}&\overline{k}\\ x_1&y_1&z_1\\ x_2&y_2&z_2\end{vmatrix}=\\[6pt] &=\overline{i}(y_1z_2-y_2z_1)-\overline{j}(x_1z_2-x_2z_1)+\overline{k}(x_1y_2-x_2y_1).\end{aligned}

I.5 (a) Area of a Triangle

The area of the triangle built on the vectors $\overline{a}=\{x_1,y_1,z_1\}$ and $\overline{b}=\{x_2,y_2,z_2\}$ is computed by the formula

\begin{aligned}S_\Delta&=\tfrac12\,\bigl|[\overline{a},\overline{b}]\bigr|=\tfrac12\left|\begin{vmatrix}\overline{i}&\overline{j}&\overline{k}\\ x_1&y_1&z_1\\ x_2&y_2&z_2\end{vmatrix}\right|=\\[6pt] &=\tfrac12\bigl|(y_1z_2-y_2z_1)\overline{i}+(x_2z_1-x_1z_2)\overline{j}+(x_1y_2-x_2y_1)\overline{k}\bigr|=\\[6pt] &=\tfrac12\sqrt{(y_1z_2-y_2z_1)^2+(x_2z_1-x_1z_2)^2+(x_1y_2-x_2y_1)^2}.\end{aligned}

I.5 (b) Vector Product of Two Combinations · Area of a Parallelogram

If $\overline{a}=\alpha\overline{m}+\beta\overline{n}$ and $\overline{b}=l\overline{m}+d\overline{n}$ , where $\alpha,\beta,l,d=\text{const}$ , then

\begin{aligned}[\overline{a},\overline{b}]&=[\alpha\overline{m}+\beta\overline{n},\ l\overline{m}+d\overline{n}]=\\[6pt] &=\alpha l[\overline{m},\overline{m}]+\alpha d[\overline{m},\overline{n}]+\beta l[\overline{n},\overline{m}]+\beta d[\overline{n},\overline{n}]=(\beta l-\alpha d)[\overline{n},\overline{m}],\end{aligned}

since $[\overline{m},\overline{n}]=-[\overline{n},\overline{m}]$ ; $[\overline{m},\overline{m}]=0$ ; $[\overline{n},\overline{n}]=0$ . The area of the parallelogram is

\begin{aligned}S_{\text{par}}&=\bigl|[\overline{a},\overline{b}]\bigr|=\bigl|(\beta l-\alpha d)[\overline{n},\overline{m}]\bigr|=\\[6pt] &=|\beta l-\alpha d|\cdot|\overline{n}|\cdot|\overline{m}|\sin\bigl(\widehat{\overline{n},\overline{m}}\bigr).\end{aligned}

I.5 (c) A Vector Perpendicular to Two Vectors

The vector product $[\overline{a},\overline{b}]$ is perpendicular to both $\overline{a}$ and $\overline{b}$ . If $\overline{a}$ and $\overline{b}$ are non-parallel, the vectors perpendicular to both form a line: every such $\overline{c}$ satisfies $\overline{c}=\lambda[\overline{a},\overline{b}]$ . For $\overline{a}=\{x_1,y_1,z_1\}$ and $\overline{b}=\{x_2,y_2,z_2\}$ , one such vector is

\begin{aligned}[\overline{a},\overline{b}]&=\begin{vmatrix}\overline{i}&\overline{j}&\overline{k}\\ x_1&y_1&z_1\\ x_2&y_2&z_2\end{vmatrix}=\\[6pt] &=(y_1z_2-y_2z_1)\overline{i}+(x_2z_1-z_2x_1)\overline{j}+(x_1y_2-x_2y_1)\overline{k},\end{aligned}

or

\overline{c}=X\overline{i}+Y\overline{j}+Z\overline{k},

where

X=y_1z_2-y_2z_1;\qquad Y=x_2z_1-z_2x_1;\qquad Z=x_1y_2-x_2y_1.

When $\overline{a}\times\overline{b}\neq\overline{0}$ , the two unit vectors perpendicular to both $\overline{a}$ and $\overline{b}$ are $\pm\dfrac{[\overline{a},\overline{b}]}{\bigl|[\overline{a},\overline{b}]\bigr|}$ .

I.6 Mixed Product of Three Vectors

Definition. The mixed product of vectors $\overline{a},\overline{b}$ and $\overline{c}$ is the scalar product of the vector $\overline{a}\times\overline{b}$ with the vector $\overline{c}$ , i.e. $(\overline{a}\times\overline{b})\cdot\overline{c}$ — also called the scalar triple product.

Properties of the mixed product:

The mixed product equals zero if: (a) at least one of the factors is zero; (b) two of the factors are collinear (lie on parallel lines); (c) the three nonzero vectors are parallel to one and the same plane (coplanarity).
The mixed product is unchanged when the dot and the cross are interchanged (without reordering the vectors): $(\overline{a}\times\overline{b})\cdot\overline{c}=\overline{a}\cdot(\overline{b}\times\overline{c})$ ; it is therefore written simply $\overline{a}\,\overline{b}\,\overline{c}$ .
It does not change under a cyclic permutation of the vectors: $\overline{a}\,\overline{b}\,\overline{c}=\overline{b}\,\overline{c}\,\overline{a}=\overline{c}\,\overline{a}\,\overline{b}$ .
Swapping any two vectors changes only its sign: $\overline{b}\,\overline{a}\,\overline{c}=-\overline{a}\,\overline{b}\,\overline{c};\ \ \overline{c}\,\overline{b}\,\overline{a}=-\overline{a}\,\overline{b}\,\overline{c};\ \ \overline{a}\,\overline{c}\,\overline{b}=-\overline{a}\,\overline{b}\,\overline{c}.$

If $\overline{a}=\{x_1,y_1,z_1\}$ ; $\overline{b}=\{x_2,y_2,z_2\}$ ; $\overline{c}=\{x_3,y_3,z_3\}$ , then

\overline{a}\,\overline{b}\,\overline{c}=\begin{vmatrix}x_1&y_1&z_1\\ x_2&y_2&z_2\\ x_3&y_3&z_3\end{vmatrix}.

From these properties it follows that:

(a) the necessary and sufficient condition for the coplanarity of three vectors is $\overline{a}\,\overline{b}\,\overline{c}=0$ , i.e.

\begin{vmatrix}x_1&y_1&z_1\\ x_2&y_2&z_2\\ x_3&y_3&z_3\end{vmatrix}=0;

(b) the volume $V_1$ of the parallelepiped built on $\overline{a},\overline{b},\overline{c}$ and the volume $V_2$ of the triangular pyramid (tetrahedron) they form are found by the formulas $V_1=\bigl|\overline{a}\,\overline{b}\,\overline{c}\bigr|$ , or

V_1=\text{abs.val}\begin{vmatrix}x_1&y_1&z_1\\ x_2&y_2&z_2\\ x_3&y_3&z_3\end{vmatrix},\qquad V_2=\tfrac16 V_1=\tfrac16\bigl|\overline{a}\,\overline{b}\,\overline{c}\bigr|,

or

V_2=\tfrac16\,\text{abs.val}\begin{vmatrix}x_1&y_1&z_1\\ x_2&y_2&z_2\\ x_3&y_3&z_3\end{vmatrix}.

I.7 Scalar and Vector Projections

The projection of a vector $\overline{AB}$ onto an axis $l$ is the value of the directed segment $A'B'$ enclosed between the projections of the start and the end of $\overline{AB}$ , taken with a positive sign when $\overline{A'B'}$ has the direction of the unit vector of the axis $l$ , and with a negative sign when $\overline{A'B'}$ and that unit vector have opposite directions. Let $\overline{l}$ be any nonzero vector along the axis $l$ ; then

\text{pr}_l\,\overline{AB}=|\overline{AB}|\cos\bigl(\widehat{\overline{AB},\,l}\bigr)=\frac{(\overline{AB}\cdot\overline{l})}{|\overline{l}|}\qquad(\overline{l}\neq\overline{0});

\overline{A'B'}=\frac{\overline{l}}{|\overline{l}|}\cdot\text{pr}_l\,\overline{AB}=\overline{l}\,\frac{(\overline{AB}\cdot\overline{l})}{|\overline{l}|^2}.

Projection of vector AB onto axis l, between the feet A′ and B′ of the perpendiculars