An Introduction to Vectors

The Definition of a Vector and Vector Terminology.

• Naively, a vector is a quantity with both a magnitude and a direction. More specifically:

  • Geometrically : A vector is an arrow whose length is the magnitude, and whose direction is given by the direction in which the arrow points.
  • Algebraically : A vector \(\vec{v}\) is an ordered \(n\) -tuple \(\vec{v}= (v_{1},\dots ,v_{n})\) of real numbers. The numbers \(v_{1},\dots ,v_{n}\) are called the components or coordinates of \(\vec{v}\text{.}\) To differentiate them from points, we sometimes use the notation \(\vec{v}= \langle v_{1},\dots ,v_{n}\rangle\text{.}\) In particular, the interactive exercises in this worksheet will not accept the \((v_{1},\dots ,v_{n})\) notation for vectors. To input a vector you will need to use the notation < v1,v2,... ,vn >.
• Algebraic and Geometric Relationship : If we place an arrow representing a vector with tail at the origin, the algebraic representation \((v_{1},\dots ,v_{n})\) gives coordinates of the head of the arrow.
• The zero vector is the vector \(\vec{0}=(0,0,\dots ,0 )\) in \(n\)-space.
• A position vector is a vector \(\vec{v}\) with its tail placed at the origin.
• The geometric representation of the vector \(\vec{v} =(v_{1},\dots ,v_{n})\) at point \(P=(a_1,\dots ,a_n)\) is the vector \(\vec{v}\) placed with its tail at \(P\) in \(n\)-space. The head of this vector lies at the point \(B=(a_1+v_1,\dots ,a_n+v_n)\)
• Vectors between points : If \(P=(a_1,\dots ,a_n)\) and \(Q=(b_1,\dots ,b_n)\) are two points, the vector \(\vec{PQ}\) is the vector between them. Its algebraic representation is \((b_1-a_1, b_2-a_2,\dots ,b_n-a_n)\text{.}\)
• A vector of length \(1\) is called a unit vector . If \(\vec{v} \neq \vec{0}\) , the unit vector pointing in the same direction as \(\vec{v}\) is \(\frac{1}{||\vec{v} ||} \vec{v}\text{.}\) The process of finding such a vector is called normalizing \(\vec{v}\text{.}\)

Operations on Vectors.

• The magnitude of \(\vec{v}=(v_{1},v_{2},\dots ,v_n)\) denoted \(|| \vec{v} ||\) is calculated as \(|| \vec{v} || =\sqrt{v_{1}^2+v_{2}^2+\dots v_n^2}\)

• Vector Addition

  • Geometrically, the sum \(\vec{v}+\vec{w}\) is a vector whose endpoint is reached after moving along \(\vec{v}\) followed by \(\vec{w}\) from the initial point.

    

  • Algebraically, if \(\vec{v}=(v_{1},v_{2},\dots ,v_n)\) and \(\vec{w}=(w_{1},w_{2},\dots ,w_n)\) , then

    \begin{equation*} \vec{v}+\vec{w}=(v_1+w_{1},v_2+w_{2},\dots ,v_n+w_n). \end{equation*}
  • Note that, \(\vec v+\vec w = \vec w+\vec v\text{.}\) Geometrically, this means that we can move first along \(\vec w\) and then along \(\vec v\) and end up at the same point (see the figure).

• Scalar Multiplication

  • Geometrically, for a scalar \(C\neq 0\text{,}\) the scalar multiple \(C\vec{v}\) is a vector with magnitude \(|C|||\vec{v}||\) and direction the same as \(\vec{v}\) if \(C\) is positive and opposite to \(\vec{v}\) if \(C\) is negative. If \(C=0\) or \(\vec{v}=0\) then \(C\vec{v}=\vec{0}\)

    

  • Algebraically, if \(\vec{v}=(v_{1},v_{2},\dots ,v_n)\) then

    \begin{equation*} C\vec{v}=(Cv_1,Cv_2,\dots ,Cv_n). \end{equation*}

• Vector Subtraction

  • Geometrically \(\vec{w} -\vec{v}\) is defined as \(\vec{w} +(-\vec{v})\text{,}\) so we move along \(\vec{w}\) and then backward along \(\vec{v}\)

    

  • Algebraically, if \(\vec{v}=(v_{1},v_{2},\dots ,v_n)\) and \(\vec{w}=(w_{1},w_{2},\dots ,w_n)\) then

    \begin{equation*} \vec{w}-\vec{v}=(w_1-v_1,w_2-v_2,\dots ,w_n-v_n) \end{equation*}

Parallel vectors.

Other Representations of Vectors.

• To differentiate them from points, we sometimes use the notation \(\vec{v}= \langle v_{1},\dots ,v_{n}\rangle\) instead of \(( v_{1},\dots ,v_{n})\text{.}\)

• The standard basis vectors for \(3\)-space are \(\vec{i}\text{,}\) \(\vec{j}\text{,}\) and \(\vec{k}\text{,}\) the unit vectors pointing in the direction of the positive \(x\)-axis, \(y\)-axis, and \(z\)-axis respectively. The vector \(\vec{v} =( a,b,c)\) can also be written as \(a\vec{i} +b\vec{j} +c\vec{k}\text{.}\) In this worksheet you can input vectors in \(3\)-space using the notation a*i+b*j+c*k or the notation <a,b,c>. The notation (a,b,c) will not be accepted for vectors (it will be interpreted as a point).
• If a vector \(\vec v\) makes an angle \(\theta\) with the positive \(x\)-axis, then \(\vec{v}=||\vec{v}|| (\cos{(\theta )} \vec{i} +\sin{(\theta )} \vec{j} )\text{.}\) We call this the polar form of \(\vec v\text{.}\)