Some Notes on Vectors

Comp.145  October 1996  Dr. A.J. Mestel

(1) A vector is a quantity with both magnitude and direction. In contrast, a scalar is just a number, with magnitude only. Usually we represent a vector in terms of its components with respect to cartesian axes, either as a row vector,

\underline{a}^{T} = (a_{1}, a_{2}, a_{3}) , or a column vector \underline{a} = ( \begin{matrix}a_{1} \\ a_{2} \\ a_{3} \end{matrix} ) .

The suffix ^{T}, denotes Transpose (see later) and is often left out.

(2) The dimension of a vector is the number of components it has. On this sheet we consider mainly 3-dimensional vectors, which we relate to ordinary three-dimensional space. The vector \underline{a} can be thought of as pointing from the origin to the point A whose cartesian coordinates are (a_{1}, a_{2}, a_{3}) . Such a vector is called the position vector of A.

(3) To add two vectors, we simply add the corresponding components:

\underline{a} + \underline{b} = (a_{1} + b_{1}, a_{2} + b_{2}, a_{3} + b_{3})^{T} .

Similarly, we can subtract two vectors. Geometrically, we can represent \underline{a}, \underline{b} and \underline{a} + \underline{b} as the sides of a triangle. The zero vector, \underline{0} = \underline{a} − \underline{a} = (0, 0, 0)^{T}.

To multiply a vector by a scalar, λ, we just multiply each component in turn:

λ\underline{a} = (λa_{1}, λa_{2}, λa_{3})^{T} .

(4) The magnitude, length, or modulus of a vector \underline{a} is written |\underline{a}| and pronounced “mod a”. In terms of the components of \underline{a},

|\underline{a}| = \sqrt{a_{1}^{2} + a_{2}^{2} + a_{3}^{2}} .

A vector \underline{a} for which |\underline{a}| = 1 is called a unit vector.

(5) The scalar product or dot product of two vectors \underline{a} and \underline{b} is written \underline{a} · \underline{b}

(or sometimes \underline{a}^{T} \underline{b} – see later) and pronounced “a dot b.” It is a scalar, given by

\underline{a} · \underline{b} = a_{1}b_{1} + a_{2}b_{2} + a_{3}b_{3} .  Alternatively,  \underline{a} · \underline{b} =|\underline{a}||\underline{b}| cos θ ,

where θ is the angle between the two vectors \underline{a} and \underline{b}. Note, for all \underline{a}, \underline{b} and \underline{c},

\underline{a} · \underline{b} = \underline{b} · \underline{a},  (\underline{a} + \underline{b}) · \underline{c} = \underline{a} · \underline{c} + \underline{b} · \underline{c},  \underline{a} · \underline{a} =|\underline{a}|^{2} .

(6) Two vectors are perpendicular if and only if \underline{a} · \underline{b} = 0. In general the angle θ between two vectors may be found from their components by using

cos θ = \frac{\underline{a} · \underline{b}}{|\underline{a}||\underline{b}|} .

(7) The vector product, or cross product of \underline{a} and \underline{b} is written \underline{a} ∧ \underline{b} or \underline{a} × \underline{b} and pronounced “a cross b”. It is defined only for 3-dimensional vectors. It is a vector perpendicular to both \underline{a} and \underline{b}, so that

\underline{a} · (\underline{a} ∧ \underline{b}) = \underline{b} · (\underline{a} ∧ \underline{b}) = 0 .

Its modulus is given by

|\underline{a} ∧ \underline{b}| = |\underline{a}||\underline{b}| sin θ ,

where θ is the angle between \underline{a} and \underline{b}. In terms of components,

\underline{a} ∧ \underline{b} = (a_{2}b_{3} − a_{3}b_{2}, a_{3}b_{1} − a_{1}b_{3}, a_{1}b_{2} − a_{2}b_{1})^{T} .

Note that for all vectors \underline{a}, \underline{b} and \underline{c}

\underline{a} ∧ \underline{b} = −\underline{b} ∧ \underline{a},  \underline{a} ∧ \underline{a} = 0,  \underline{a} ∧ (\underline{b} + \underline{c}) = \underline{a} ∧ \underline{b} + \underline{a} ∧ \underline{c} .

Two non-zero vectors \underline{a} and \underline{b} are parallel (that is, they point in the same or opposite directions) if and only if \underline{a} ∧ \underline{b} = 0.

(8) A quantity such as (\underline{a} ∧ \underline{b}) · \underline{c} is called a scalar triple product. It is equal to zero if and only if the three vectors \underline{a}, \underline{b} and \underline{c} lie in the same plane, in which case they are said to be linearly dependent. Otherwise, they are linearly independent. By the end of the course, you will know what a determinant is, and will understand the formula

(\underline{a} ∧ \underline{b}) · \underline{c} = | \begin{matrix}a_{1} \\ b_{1} \\ c_{1} \end{matrix} \begin{matrix}a_{2} \\ b_{2} \\ c_{2} \end{matrix} \begin{matrix}a_{3} \\ b_{3} \\ c_{3} \end{matrix} | .

It can be shown that (\underline{a} ∧ \underline{b}) · \underline{c} = \underline{a} · (\underline{b} ∧ \underline{c}) and (\underline{a} ∧ \underline{b}) · \underline{c} = (\underline{b} ∧ \underline{c}) · \underline{a} = (\underline{c} ∧ \underline{a}) · \underline{b}.

(9) Various three-dimensional shapes and curves may be simply represented by vector equations. In what follows \underline{R} = (x, y, z)^{T} is the position vector of a variable point. A sphere whose centre has position vector \underline{a} and whose radius is k can be described by

|\underline{R} − \underline{a}| = k .

(10) The equation of a plane perpendicular to a vector \underline{n} can be written

\underline{R} · \underline{n} = d .

If \underline{n} is a unit vector, then d is the perpendicular distance from the origin to the plane.

(11) A straight line passing through the point with position vector \underline{a} in the direction of \underline{c} takes the form:

\underline{R} = \underline{a} + μ \underline{c} ,

where μ is a parameter which may take any value. Thus, the line passing through the points with position vectors \underline{a} and \underline{b} is

\underline{R} = \underline{a} + μ (\underline{b} − \underline{a})  or  \underline{R} = (1 − μ)\underline{a} + μ \underline{b} .

The point midway between \underline{a} and \underline{b} occurs when μ = \frac{1}{2} and \underline{R} = \frac{1}{2}(\underline{a} + \underline{b}).