# 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.

## Representation on a Coordinate Plane

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 \( \vec{V} \).

We have

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

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.

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.)

## Three-Dimensional Vectors

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

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

in which case the length becomes

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

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

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

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 \(3\vec V\).

This can be done in two ways:

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

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

## See Also

## Notes

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