5. Vectors

c. Scalar Multiplication

In discussing vectors, we normally specify a vector as an ordered pair of components. However, there is a second notation which involves two special vectors: \(\hat\imath\) and \(\hat\jmath\).

5. \(\hat\imath\hat\jmath\) Notation

What happens if we take an arbitrary linear combination of \(\hat\imath\) and \(\hat\jmath\) with coefficients \(a\) and \(b\)? \[\begin{aligned} a\hat\imath+b\hat\jmath &=a\left\langle1,0\right\rangle+b\left\langle0,1\right\rangle \\ &=\left\langle a,0\right\rangle+\left\langle0,b\right\rangle =\left\langle a,b\right\rangle \end{aligned}\] We get the vector \(\left\langle a,b\right\rangle\). Turning this around, we can express any vector \(\vec v\) in terms of \(\hat\imath\) and \(\hat\jmath\): \[ \vec v=\left\langle v_1,v_2\right\rangle =v_1\hat\imath+v_2\hat\jmath \]

For example, the vector \(\vec v=\left\langle3,-2\right\rangle\) can be written as \(\vec v=3\hat\imath-2\hat\jmath\).

The ordered pair notation, \(\vec v=\left\langle v_1,v_2\right\rangle\), and the \(\hat\imath\,\hat\jmath\) notation, \(\vec v=v_1\hat\imath+v_2\hat\jmath\), can be used interchangably. For some purposes one or the other may be more useful.

You need to be able to go back and forth as needed.