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Question: How do you compute the dot product for \(u=3i+2j\) and \(v=-2i-3j\)?...

How do you compute the dot product for u=3i+2ju=3i+2j and v=−2i−3jv=-2i-3j?

Explanation

Solution

We have two vectors namely u,vu,v with i∧ \overset{\wedge }{\mathop{i}}\, and j∧ \overset{\wedge }{\mathop{j}}\, as unit vectors. Dot product of the two vectors involves multiplying the i∧ \overset{\wedge }{\mathop{i}}\,component of both the vectors together, that is, 3.(−2)3.(-2) and multiplying the j∧ \overset{\wedge }{\mathop{j}}\, component of both the vectors , that is, 2.(−3)2.(-3) and adding them up. Hence, we get the dot product of both the vectors.

Complete step by step solution:
According to the given question, we have been given two vectors uu and vv, on which we have to compute the dot product.
Dot product, is also called the scalar product, as it involves multiplication of similar components to give a scalar result (a number).
Also,
i∧ .i∧ =1\overset{\wedge }{\mathop{i}}\,.\overset{\wedge }{\mathop{i}}\,=1
j∧ .j∧ =1\overset{\wedge }{\mathop{j}}\,.\overset{\wedge }{\mathop{j}}\,=1

The vectors given to us is,
u=3i+2ju=3i+2j
v=−2i−3jv=-2i-3j
So the dot product of the two vectors is,
u.v=(3i+2j).(−2i−3j)u.v=(3i+2j).(-2i-3j)-----(1)
Here, we will multiply the i∧ \overset{\wedge }{\mathop{i}}\,component of both the vectors together, similarly, we will multiply the j∧ \overset{\wedge }{\mathop{j}}\, component of both the vectors together, we get,
⇒u.v=3.(−2)+2.(−3)\Rightarrow u.v=3.(-2)+2.(-3)-----(2)
We will multiply the terms in the equation (2). The first term is 3 multiplied by (-2) which gives us -6. And the second term has 2 multiplied by (-3) which also gives us -6.
So, we have,
⇒u.v=−6+(−6)\Rightarrow u.v=-6+(-6)
Now, we have two (-6) to be added up. We get,
⇒u.v=−6−6\Rightarrow u.v=-6-6
⇒u.v=−12\Rightarrow u.v=-12
Therefore, the dot product of the given vectors is −12-12.

Note: The dot product involves multiplication of an i∧ \overset{\wedge }{\mathop{i}}\, component with another i∧ \overset{\wedge }{\mathop{i}}\, component and is not multiplied with all the components. We saw in the above solution how a dot product works. We also have an operation called the cross product. It follows a different set of procedures to carry out the operation.
Also,
i∧ ×i∧ =0\overset{\wedge }{\mathop{i}}\,\times \overset{\wedge }{\mathop{i}}\,=0
j∧ ×j∧ =0\overset{\wedge }{\mathop{j}}\,\times \overset{\wedge }{\mathop{j}}\,=0