Vectors

Most quantities with which you are probably familiar are represented by single numbers: \( \frac{\sqrt{2}}{3} \), 15 kilograms, or 25.63 seconds, for instance. Such quantities are known as scalar quantities or simply scalars.

However, some quantities have both a size and direction and thus require two or more numbers to specify completely. Such quantities are known as vector quantities or vectors.\(^\text{[1]}\) For instance, while the speed of an object is simply its rate of motion \( (18 \, \text{km}/\text{h}), \) an object's velocity reflects not just its rate of motion but also the direction of its motion \( (18 \, \text{km}/\text{h} \) southwest\().\) As such, speed is a scalar quantity while velocity is a vector quantity. Other vector quantities include force, displacement, and electric field.

Many important physical and mathematical quantities are vectors, and the analysis of generalized vectors and their properties (a subject known as linear algebra) forms part of the core of modern mathematics.

Before going further, we first need to know what a vector is.

Mathematically, a directed line segment is called a vector. Or, in other words, a line segment with a specified magnitude and direction is called a vector.

One elementary way to specify a vector is simply to give its size and direction \((\)e.g., \(5 \, \text{meters} \) north or \( 12 \) at \( 45^\circ )\). However, in some cases, neither quantity may be immediately apparent. Sometimes it is easier to specify two points—an initial point \( A \) and a terminal point \( B \)—and represent a vector as the (directed) displacement from one to the other.

On a Cartesian plane, one can take \( A = (x_A, y_A) \) and \( B = (x_B, y_B) \), in which case the horizontal part of the displacement is \( x_B - x _A \) and the vertical part is \( y_B - y_A \). Generally, one notates a vector \( \vec{V} \) by indicating both parts, also called components, in the form of an ordered pair

\[ \vec{V} = (x_B - x_A, y_B - y_A), \]

or simply

\[ \vec{V} = (v_x, v_y) \]

with \( v_x = x_B - x_A \) and \( v_y = y_B - y_A \).

To differentiate a vector from a scalar, one generally writes a vector with an arrow, as shown (although this notation is not strictly required).

In general, one is only interested in the displacement—that is to say, one does not differentiate between vectors with different initial and terminal points as long as each component is the same. Vectors with identical components are considered equal or congruent. (Indicating the initial and terminal points is simply a visual and calculation aid.)

A vector \( \vec{V} \) has initial point \( A = (-1,2) \) and terminal point \( B = (3,2) \). Determine the components of \( \vec{V} \).

ANSWER

We have

\[ \vec{V} = \big(3- (-1), 2 - 2\big) = (4, 0).\ _\square \]

The size, length, or magnitude of a vector is usually taken as the Euclidean distance given by the Pythagorean theorem and written as \( | \vec{V} | \):

\[ \vec{V} = \sqrt{(x_B - x_A)^2 + (y_B - y_A)^2} = \sqrt{v_x^2 + v_y^2}. \]

The direction is often given as the angle \( \theta \) with respect to the positive \( x \)-axis, which is given by

\[ \tan{\theta} = \frac{y_B - y_A}{x_B - x_A} = \frac{v_y}{v_x}. \]

A certain vector \(\vec{V}\) can be represented as \(6\hat{\imath}+8\hat{\jmath}\) . Find its magnitude and the angle it makes with respect to the positive \(x\)-axis.

ANSWER

The magnitude is

\[\Big|\vec{V}\Big|=\sqrt{v_x^2+v_y^2}=\sqrt{6^2+8^2}=\sqrt{36+64}=\sqrt{100}=10,\]

and the angle is

\[\tan(\theta_{\text{w.r.t. } x\text{-axis}})=\dfrac{v_y}{v_x}=\dfrac{8}{6}=\dfrac{4}{3} \implies \theta_{\text{w.r.t. } x\text{-axis}}=\tan^{-1}\left(\dfrac{4}{3}\right).\ _\square\]

Plane vectors can be specified both by indicating both components or by giving the magnitude and angle. In both cases, the two values are sufficient to provide all information about the vector and move between both the "component" and "magnitude and angle" representations. (Equivalently, one can think of the "magnitude and angle" representation as simply the coordinates of a point in plane polar coordinates.)

In three dimensions, one adds a third component for the \( z \)-axis:

\[ \vec{V} = (v_x, v_y, v_z). \]

If \(\vec{V} = (v_x, v_y, v_z)\) is any point in 3D space, then the vector \(\vec{OV}\) having origin \(O(0,0,0)\) and \(\vec{V}\) as its initial and terminal points, respectively, is called the position vector of \(\vec{V}\) and its length is given by

\[ \Big|\vec{V}\Big| = \sqrt{v_x^2 + v_y^2 + v_z^2}. \]

A vector \( \vec{V} \) has initial point \( A = (-1,2,5) \) and terminal point \( B = (3,2,2) \). Find the length of \( \vec{V} \).

ANSWER

We have

\[ \Big|\vec{V}\Big| = \sqrt{\big(3-(-1)\big)^2 + (2 - 2)^2 + (2 - 5)^2} = 5.\ _\square \]

One may "scale" a vector by multiplying it by a scalar value. Such a product \(\big(\)between a scalar \( a \) and a vector \( \vec{V}\big) \) is written as \( a\vec{V} = a(v_x, v_y, v_z) \) and defined as

\[ a\vec{V} = (av_x, av_y, av_z ). \]

Such a product merely multiplies the length by a factor \( a \), as

\[ \Big|a\vec{V}\Big| = \sqrt{(av_x)^2 + (av_y)^2 + (av_z)^2} = \sqrt{a^2\big(v_x^2 + v_y^2 + v_z^2\big)} = a \sqrt{v_x^2 + v_y^2 + v_z^2} = a \Big|\vec{V}\Big|. \]

A vector \(\vec V=(3,4,12)\). Find the length of \(3\vec V\).

ANSWER

This can be done in two ways:

• The first way: \(\Big|3\vec V\Big|=\big|3\times(3,4,12)\big|=\big|(9,12,36)\big|=\sqrt{9^2+12^2+36^2}=39.\)

• The second way: \(\Big|3\vec V\Big|=3\Big|\vec V\Big|=3\big|(3,4,12)\big|=3\sqrt{3^2+4^2+12^2}=3\times 13=39.\) \(_\square\)

Note : The vector \(\vec{AB} = \vec{OB} - \vec{OA},\) where \(\vec{OA}, \vec{OB}\) are the position vectors of points \(A\) and \(B,\) respectively.

The following are various types of vectors:

• Null vector: A vector whose initial and terminal points are the same is called a null vector \(\big(\vec{0}\big)\) or zero vector. Its magnitude is zero and its direction is indeterminate.

• Unit vector: A vector whose magnitude is unity (1 unit) is called a unit vector. If \(\vec{a}\) is any vector, then its unit vector is denoted by \(\hat{a}\) and the unit vector of \(\vec{a}\) is given by \(\hat{a} = \dfrac{\vec{a}}{|\vec{a}|}\).

• Equal vector: Two vectors \(\vec{a}\) and \(\vec{b}\) are said to be equal if they have the same magnitude and direction.

• Negative of a vector: A vector whose magnitude is the same as that of the given vector but in opposite direction to it is called the negative of that vector. If \(\vec{a}\) is any vector, then its negative is denoted by \(-\vec{a}\). Also, if \(\vec{AB}\) is any vector, then its negative is given by \(\vec{BA}\). Always remember that \(\vec{AB} = - \vec{BA}\).

• Parallel vector: Two vectors are said to be parallel if and only if they have the same support or parallel support. In other words, if \(\vec{a}\) and \(\vec{b}\) are any two parallel vectors, then \(\vec{a} = \alpha \vec{b},\) where \(\alpha\) is some scalar constant. If the two vectors have the same direction, they are said to be parallel vectors (or like vectors); but if the two vectors are in opposite directions, they are said to be anti-parallel vectors (or unlike vectors).

• Co-linear vector: Two vectors \(\vec{a}\) and \(\vec{b}\) are said to be co-linear if they have the same direction, parallel or anti-parallel.

• Co-planar vector: Vectors whose supports are in the same plane or parallel to the same plane are called co-planar vectors. Vectors which are not co-planar are called non-co-planar vectors.

• Co-initial vector: Two or more vectors having the same initial point are called co-initial vectors.

Let \(a_1,a_2,a_3, \cdots , a_n\) be vectors and \(x_1,x_2,x_3, \cdots , x_n\) be scalars. Then the vector \(x_1a_1 + x_2a_2 + x_3a_3 + \cdots + x_na_n\) is called a linear combination of the vectors \(a_1,a_2,a_3, \cdots , a_n\).

• \(2a - b + 3c\) is a linear combination of \(a,b,c\).
• Three vectors are co-planar if and only if one of them is a linear combination of the other two.

• Vector Addition

• Unit Vectors

• Dot Product

• Cross Product

[1] Strictly speaking, in this article we refer to geometric or Euclidean vectors. Generalized vectors need not be multi-dimensional.