Question

Calculate the dot product of a= (22, 2, 7) & b= (12, -9, 11). And Cross Product of c= (4, 0, 3) & d= (3, 1, 7).

Answer 1

1) Use the component formula for the dot product of three-dimensional vectors

$$\bold{a\cdot b}=a_1 b_1 +a_2 b_2+a_3b_3$$ $$\bold{a\cdot b}=22\cdot 12+2\cdot(-9)+7\cdot 11=264-18+77=323$$

2) Use the formula for the cross product of three-dimensional vectors

$$\bold{a\times b}=\begin{vmatrix} a_2 & a_3 \\ b_2 & b_3 \end{vmatrix}\bold{i}-\begin{vmatrix} a_1 & a_3 \\ b_1 & b_3 \end{vmatrix}\bold{j}+\begin{vmatrix} a_1 & a_2 \\ b_1 & b_2 \end{vmatrix}\bold{k}$$

$$\bold{a\times b}=\begin{vmatrix} 2 & 7 \\ -9 & 11 \end{vmatrix}\bold{i}-\begin{vmatrix} 22 & 7 \\ 12 & 11 \end{vmatrix}\bold{j}+\begin{vmatrix} 22 & 2 \\ 12 &-9 \end{vmatrix}\bold{k}$$

$$\bold{a\times b}=(22+63)\bold{i}-(242-84)\bold{j}+(-198-24)\bold{k}$$

$$\bold{a\times b}=85\bold{i}-158\bold{j}-222\bold{k}$$

$$\bold{a\times b}=(85,-158,-222)$$

Answer:

1) 323

2) (85,-158,-222)