Question

Determine the dot product of the vector\(a\)and\(b\)and verify\(a \times b\) is orthogonal on both\(a\)and\(b.\)

Short answer

The cross product of the vector\(a\)and\(b\) is\(16{\rm{i}} + 48{\rm{k}}{\rm{.}}\)

Step-by-step solution

The cross product formula.

If the vectors\({\bf{a}} = \left\langle {{a_1},{a_2},{a_3}} \right\rangle \)and\({\bf{b}} = \left\langle {{b_1},{b_2},{b_3}} \right\rangle {\rm{,}}\)then the cross product of\({\bf{a}}\)and\({\bf{b}}\)is\({\rm{a}} \times {\rm{b}} = \left| {\begin{array}{*{20}{c}}{\rm{i}}&{\rm{j}}&{\rm{k}}\\{{a_1}}&{{a_2}}&{{a_3}}\\{{b_1}}&{{b_2}}&{{b_3}}\end{array}} \right|\).

Use the cross product formula for calculation.

The two given vectors are\({\rm{a}} = \langle 6,0, - 2\rangle \) and\({\rm{b}} = \langle 0,8,0\rangle \).

Apply the formula with\({a_1} = 6,{a_2} = 0,{a_3} = - 2,{b_1} = 0,{b_2} = 8,\) and\({b_3} = 0\)and obtain the cross product of two vectors.

\(\begin{array}{l}a \times b = \left| {\begin{array}{*{20}{c}}i&j&k\\6&0&{ - 2}\\0&8&0\end{array}} \right|\\ = \left| {\begin{array}{*{20}{c}}0&{ - 2}\\8&0\end{array}} \right|{\rm{i}} - \left| {\begin{array}{*{20}{c}}6&{ - 2}\\0&0\end{array}} \right|j + \left| {\begin{array}{*{20}{c}}6&0\\0&8\end{array}} \right|k\\ = (0 + 16){\rm{i}} - (0 - 0){\rm{j}} + (48 - 0){\rm{k}}\\ = 16i + 48k\end{array}\)

Thus, the cross product of the vector\(a\)and\(b\) is\(16{\rm{i}} + 48{\rm{k}}{\rm{.}}\)

Note that the cross product \(a \times b\) is orthogonal to both \(a\) and \(b\) if \((a \times b) \cdot a = 0\) and\((a \times b) \cdot b = 0\).

Rewrite the vector\({\rm{a}} = \langle 6,0, - 2\rangle \) in terms of unit vector as\(a = 6{\bf{i}} - 2k.\)

Calculate the vector value\((a \times b) \cdot a\) as follows.

Substitute\(a \times b = 16i + 48k\) and\(a = 6i - 2k{\rm{ in }}(a \times b) \cdot a\)and obtain the dot product as follows.

Rewrite the vector\({\rm{b}} = \langle 0,8,0\rangle \)in terms of unit vector as\({\rm{b}} = 8{\rm{j}}.\)

Calculate the vector value\((a \times b) \cdot b\) as follows.

Substitute\(a \times b = 16i + 48k\)and\(b = 8j{\rm{ in }}(a \times b) \cdot b\) and obtain the dot product as follows.

Thus, the cross product of the vector\(a\)and\(b\) is\(16{\rm{i}} + 48{\rm{k}}{\rm{.}}\)