V1: Introduction to vectors

Introducing the basic concept of vectors (which are measurable quantities with direction such as force) and the concept of a scalar (which is a quantity without a direction such as heat or thermal energy).

We cover vector components (aligned along the horizontal and vertical or the 3-dimensional axes), unit vectors (a vector of size = 1), vector magnitude (or size) and how to add and subtract vectors.

Vectors and scalars

One example of a vector is velocity . The velocity of an object is determined by the magnitude (speed) and direction of travel. Other examples of vectors are force, displacement and acceleration.

A scalar is a quantity that has magnitude only. Mass, time and volume are all examples of scalar quantities.

Vectors in three-dimensional space are defined by three mutually perpendicular directions and and can be denoted as bold letters or as in this worksheet \(\vec{a}\) or \(\vec{b}\) or \(\vec{c}.\)

A vector in the opposite direction from \(\vec{a}\) is denoted by \(-\vec{a}\).

Vectors can be added or subtracted graphically using the triangle rule.

Adding and subtracting vectors

Triangle rule:

[l-image url="https://rmitlibrary.github.io/cdn/learninglab/images/maths-v1_adding_vectors.png" alt="Adddition of vectors"]

To add vectors \(\vec{a}\) and \(\vec{b}\) shown above place the tail of vector \(\vec{b}\) at the head of vector \(\vec{a}\) (point Q).

The vector sum, \(\vec{a}+\vec{b},\) is the vector \(\overrightarrow{PR}\) , from the tail of vector \(\vec{a}\) to the head of \(\vec{b}.\)

To subtract \(\vec{b}\) from \(\vec{a}\), reverse the direction of \(\vec{b}\) to give \(-\vec{b}\) then add \(\vec{a}\) and \(-\vec{b}\).

\(\vec{a}-\vec{b}=\vec{a}+(-\vec{b})\)

Vector \(\overrightarrow{PR}\) is equal to the vector \(\vec{a}-\vec{b}\).

Components of a vector

In the diagram below the vector \(\vec{r}\) is represented by \(\overrightarrow{OP}\) where \(P\) is the point \((x,y,z)\).

if \(\vec{i}\) ,\(\vec{j}\) , and \(\vec{k}\) are vectors of magnitude one1 A vector of magnitude one is called a unit vector. Magnitude of vectors and unit vectors are discussed later in this module. parallel to the positive directions of the \(x\)- axis , \(y\)-axis and \(z\)-axis respectively, then:

\(x\vec{i}\) is a vector of length x in the direction of the \(x\)-axis

\(y\vec{i}\) is a vector of length y in the direction of the \(y\)-axis

\(z\vec{k}\) is a vector of length z in the direction of the \(z\)-axis

\(\overrightarrow{OP}\) is then the vector \(x\vec{i}+y\vec{j}+z\vec{k}\)

\(x\), \(y\) and \(z\) are called the components of the vector.

The notation \((x,y,z)\) will be used to denote the vector \(\vec{r}=(x\vec{i}+y\vec{j}+z\vec{k})\) as well as the co-ordinates of a point \(P\) \((x,y,z)\). The context will determine the correct meaning.

Vectors may also be added or subtracted by adding or subtracting their corresponding components.

Example

If \(\vec{a}=(-3,4,2)\) and \(\vec{b}=(-1,-2,3)\) , find:

\[\begin{align*} \vec{a}+\vec{b} & =(-3,4,2)+(-1,-2,3)\\ & =(-3+(-1),4+(-2),2+3)\\ & =(-4,2,5) \end{align*}\] and

\[\begin{align*} \vec{a}-\vec{b} & =(-3,4,2)-(-1,-2,3)\\ & =(-3-(-1),4-(-2),2-3)\\ & =(-2,6,-1) \end{align*}\]

See Exercise 1.

Directed line segment

The directed line segment, or geometric vector, \(\overrightarrow{PQ}\) , from\(P(x_{1},y_{1},z_{1})\) to \(Q(x_{2},y_{2},z_{2})\) is found by subtracting the co-ordinates of \(P\) (the initial point) from the co-ordinates of \(Q\) (the final point).

\(\overrightarrow{PQ}=(x_{2}-x_{1})\vec{i}+(y_{2}-y_{1})\vec{j}+(z_{2}-z_{1})\vec{k}\)

Example

\(PQ=(5-3)i+(6-4)+(-1-1)k\)

\(PQ=2i+2j-2k\)

The directed line segment \(\overrightarrow{PQ}\) is represented by the vector \(2\vec{i}+2\vec{j}-2\vec{k}\), or \((2,2,-2)\). Any other directed line segment with the same length and same direction as \(\overrightarrow{PQ}\) is also represented by \(2\vec{i}+2\vec{j}-2\vec{k}\) or \((2,2,-2)\).

The directed line segment \(\overrightarrow{QP}\) has the same length as \(\overrightarrow{PQ}\) but is in the opposite direction.

\[\begin{alignat*}{1} \overrightarrow{QP} & =-\overrightarrow{PQ}\\ & =-(2\vec{i}+2\vec{j}-2\vec{k})\\ & =-2\vec{i}-2\vec{j}+2\vec{k}\\ is\ also\ & =(-2,-2,2) \end{alignat*}\]

Position vector

The position vector of any point is the directed line segment from the origin \(O\) \((0,0,0)\) to that point and is given by the co-ordinates of of the point.

The position vector of \(P(3,4,1)\) is \(3\vec{i}+4\vec{j}+\vec{k}\), or \((3,4,1)\).

See Exercise 2.

Magnitude of a vector

if \(\vec{a}=a_{1}\vec{i}+a_{2}\vec{j}+a_{3}\vec{k}\) the length or magnitude of \(\vec{a}\) is written as \(\left|\vec{a}\right|\) or \('\vec{a}'\) and is evaluated as:

\(\left|\vec{a}\right|=\sqrt{(a_{1})^{2}+(a_{2})^{2}+(a_{3})^{2}}\)

Example

The length of the vector \(2\vec{i}+3\vec{j}-5\vec{k}\) equals \(\sqrt{2^{2}+3^{2}+(-5)^{2}}=\sqrt{38}\)

In this case \(a_{1}=2,a_{2}=3,a_{3}=-5\)

\(a_{1},a_{2},a_{3}\) are referred to as the components of vector \(\vec{a}\).

Unit vector

Any vector with a magnitude of one is called a unit vector.

If \(\vec{a}\) is any vector then a unit vector parallel to \(\vec{a}\) is written \(\hat{a}\) (a “hat”). The “hat” symbolises a unit vector.

The unit vector \(\hat{a}\) equals vector \(\vec{a}\) divided by its magnitude \(\left|\vec{a}\right|\) .

\[\begin{alignat*}{1} \hat{a} & =\frac{\vec{a}}{\left|\vec{a}\right|}\\ Rearanging\ & \ gives\\ \vec{a} & =\left|\vec{a}\right|\hat{a} \end{alignat*}\]

Of particular importance are unit vectors \(\vec{i}\), \(\vec{j}\) and \(\vec{k}\), parallel to the \(x\) , \(y\), and \(z\) axes respectively.

Examples

1. If \(\overrightarrow{PQ}\) is the line \(2\vec{i}-5\vec{j}+\vec{k}\) find a unit vector parallel to \(\overrightarrow{PQ}\)

\(\overrightarrow{PQ}=\sqrt{2^{2}+\left(-5\right)^{2}+1^{2}}=\sqrt{30}\)

A unit vector parallel to \(\overrightarrow{PQ}\) is :

\[\begin{alignat*}{1} \overrightarrow{PQ} & =\frac{\overrightarrow{PQ}}{\left|\overrightarrow{PQ}\right|}\\ & =\frac{(2\vec{i}-5\vec{j}+\vec{k})}{\sqrt{30}}\\ & =\frac{1}{\sqrt{30}}(2\vec{i}-5\vec{j}+\vec{k}) \end{alignat*}\]

2. if \(\vec{a}\) \(=(1,2,3)\) a unit vector parallel to \(\vec{a}\) is:

\[\begin{alignat*}{1} \hat{a} & =\frac{\vec{a}}{\left|\vec{a}\right|}\\ & =\frac{(1i+2j+3k)}{\sqrt{1^{2}+2^{2}+3^{2}}}\\ & =\frac{(1i+2j+3k)}{\sqrt{14}}\\ & =\frac{1}{\sqrt{14}}(1i+2j+3k) \end{alignat*}\]

see Exercise 3.

Multiplication by a scalar

To multiply a vector \(\vec{a}=\) \(a_{1}\vec{i}+a_{2}\vec{j}+a_{3}\vec{k}\) by a scalar, \(m\), multiply each component of \(\vec{a}\) by \(m\).

\[\begin{alignat*}{1} m\vec{a} & =ma_{1}\vec{i}+ma_{2}\vec{j}+ma_{3}\vec{k} \end{alignat*}\]

The result is a vector of length \(m\times\left|\vec{a}\right|\)

If \(m>0\) the resultant vector is in the same direction as \(\vec{a}\)

If \(m<0\) the resultant vector is in the opposite direction from \(\vec{a}\).

Two vectors a and b are said to be parallel if and only if \(\vec{a}=k\vec{b}\) where \(k\) is a real constant.

Example

Multiply \(\vec{a}=(3\vec{i}+\vec{j}-2\vec{k})\) by \(7\) and show \(\left|7\vec{a}\right|=7\left|\vec{a}\right|\)

\[\begin{alignat*}{1} 7\vec{a} & =7(3\vec{i}+\vec{j}-2\vec{k})\\ & =21\vec{i}+7\vec{j}-14\vec{k}\\ so\ that\\ \left|7\vec{a}\right| & =\sqrt{21^{2}+7^{2}+(-14)^{2}}\\ & =\sqrt{686}=\sqrt{49\times14}\\ & =7\sqrt{14} \end{alignat*}\]

Also the magnitude of \(\vec{a}\) is \[\begin{alignat*}{1} \left|\vec{a}\right| & =\sqrt{3^{2}+1^{2}+(-2)^{2}}=\sqrt{14}\\ 7\left|\vec{a}\right| & =7\sqrt{14} \end{alignat*}\]

Therefore we can derive:

\[\begin{alignat*}{1} \left|7\vec{a}\right| & =7\left|\vec{a}\right|=7\sqrt{14}. \end{alignat*}\]

See Exercise 4.

Exercise 1

Given \(\vec{a}=(2,1,1)\) , \(\vec{b}=(1,3,-3)\) and \(\vec{c}=(0,3,-2)\) find:

1. \(\vec{a}+\vec{b}\)
2. \(\vec{a}+\vec{c}\)
3. \(\vec{c}-\vec{b}\)
4. \(\vec{a}-\vec{b}\).

Exercise 2

1. Given the points \(A(3,0,4)\) , \(B(-2,4,3)\), and \(C(1,-5,0)\), find:

\(\text{i}.\;\overrightarrow{AB}\quad\) ii. \(\overrightarrow{AC}\quad\) iii. \(\overrightarrow{CB}\quad\) iv. \(\overrightarrow{BC}\quad\) v. \(\overrightarrow{CA}\)
Compare your answers to (ii) and (v), and also to (iii) and (iv). What do you notice?

2. What are the position vectors of the points \(A,\,B\) and \(C\)?

Exercise 3

1. Find the length of the vectors:
   \(\text{(i)}\;(3,-1,-1)\quad\) (ii) \((0,2,4)\quad\) (iii) \((0,-2,0)\)
2. Given the points \(A\,(3,0,4)\), \(B\,(0,4,3)\) and \(C\,(1,-5,0)\);
   find unit vectors parallel to \(\text{(i)}\;\overrightarrow{BA}\quad\text{(ii) $\overrightarrow{CB}\quad\text{(iii) $\overrightarrow{AC}$ .}$ }\)

Exercise 4

1. Expand the following: (i) \(3(\vec{i}+3\vec{j}-5\vec{k})\) (ii) \(-4(\vec{j}-3\vec{k})\)
2. If \(\vec{a}=(2,-2,1)\) , \(\vec{b}=(0,1,1)\) and \(\vec{c}=(-1,3,-2)\) , find (i) \((2\vec{a}+3\vec{b})\) (ii) \((3\vec{a}-2\vec{b})\) (iii) \((2\vec{a}-\vec{b}+2\vec{c})\) (iv) a unit vector parallel to \(2\vec{a}-\vec{b}\)
3. Write down a vector three times the length of \((6\vec{i}+2\vec{j}-5\vec{k})\) and in the opposite direction.

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