Question

Evaluate the dot product $\stackrel{\rightharpoonup }{A}.\stackrel{\rightharpoonup }{B}$,if

a. $\stackrel{\rightharpoonup }{A}=3\hat{i}+4\hat{j}$and $\stackrel{\rightharpoonup }{B}=2\hat{i}-6\hat{j}$

b.$\stackrel{\rightharpoonup }{A}=3\hat{i}-2\hat{j}$and$\stackrel{\rightharpoonup }{B}=6\hat{i}+4\hat{j}$

Short answer

a. $\stackrel{\rightharpoonup }{A}.\stackrel{\rightharpoonup }{B}=-18$

b.$\stackrel{\rightharpoonup }{A}.\stackrel{\rightharpoonup }{B}=10$

Step-by-step solution

Given information (Part a)

$\stackrel{\rightharpoonup }{A}=3\hat{i}+4\hat{j}$

$\stackrel{\rightharpoonup }{B}=2\hat{i}-6\hat{j}$

Dot product (part a)

$$\begin{gathered} \stackrel{\rightharpoonup }{A}.\stackrel{\rightharpoonup }{B}=(3\hat{i}+4\hat{j}).(2\hat{i}-6\hat{j}) \\ \stackrel{\rightharpoonup }{A}.\stackrel{\rightharpoonup }{B}=(3\times 2)+(4\times (-6\left)\right) \\ \stackrel{\rightharpoonup }{A}.\stackrel{\rightharpoonup }{B}=6-24=-18 \end{gathered}$$

Given information (Part b)

$\stackrel{\rightharpoonup }{A}=3\hat{i}-2\hat{j}$

$\stackrel{\rightharpoonup }{B}=6\hat{i}+4\hat{j}$

Dot product (part b)

$$\begin{gathered} \stackrel{\rightharpoonup }{A}.\stackrel{\rightharpoonup }{B}=(3\hat{i}-2\hat{j}).(6\hat{i}+4\hat{j}) \\ \stackrel{\rightharpoonup }{A}.\stackrel{\rightharpoonup }{B}=(3\times 6)+\left(\right(-2)\times 4) \\ \stackrel{\rightharpoonup }{A}.\stackrel{\rightharpoonup }{B}=18-8=10 \end{gathered}$$