Question

How do I find the dot product of vectors $v = 2i - 3j$ and $w = i - j$?

Answer 1

Here in this question, we have to find the dot product of vectors. Here I, j and k are the unit vectors. While multiplying the two vectors there is a rule for the unit vectors. By applying this rule and the arithmetic operation multiplication we can find the solution for this question.

Complete step-by-step solution:
The vectors are defined as a vector is an object which is having both a magnitude and direction. Unit vectors are vectors whose magnitude is exactly 1 unit. Specifically, the unit vectors [0,1] and [1,0] can form together any other vector. The unit vectors are I, j and k.
While multiplying the vectors have two different kinds. One is dot product and other is cross product.
The dot product for the unit vectors are

$$i.i = 1 \\\ j.j = 1 \\\ k.k = 1 \\\$$

The cross product for the unit vectors are

$$i \times j = k \\\ j \times k = i \\\ k \times i = j \\\$$

Now consider the above question $v = 2i - 3j$ and $w = i - j$
The dot product of these vectors are
$v.w = \left( {2i - 3j} \right).\;(i - j)$
On multiplying we get
$\Rightarrow v.w = 2i\;(i - j) - 3j(i - j)$
On simplifying we get
$\Rightarrow v.w = 2i\;.i - 2i.j - 3j.i + 3j.j$
If the unit vectors are same then dot product will be 1. If the unit vectors are different then the dot product will be 0. So we have
$\Rightarrow v.w = 2(1) - 2(0) - 3(0) + 3(1)$
On simplifying we get
$\Rightarrow v.w = 2 - 0 - 0 + 3$
On further simplifying we get
$\Rightarrow v.w = 5$
Hence we have simplified the given vectors.
Therefore the dot product of $v = 2i - 3j$ and $w = i - j$is $v.w = 5$

Note: In the vectors there are different forms of vectors. The i, j and k are the unit vectors. These unit vectors represent the direction. When we find the dot product the unit vectors will vanish when we find the cross product the unit vectors will not vanish and it will remain. In the cross product the unit vector will be negative for some unit vectors.