If A= <6 ,9 ,-1 >A=<6,9,−1> and B= <9 ,-1 ,8 >B=<9,−1,8>, what is A*B -||A|| ||B||A⋅B−||A||||B||?

1 Answer
Lotusbluete
Mar 23, 2016

A * B - ||A|| * ||B|| = 37 - sqrt(118)*sqrt(146)A⋅B−||A||⋅||B||=37−√118⋅√146

Explanation:

A * BA⋅B is a dot product (or scalar product) and can be computed as follows:

A = < color(blue)(6), color(green)(" "9), color(purple)(-1) >A=<6, 9,−1>
B = < color(blue)(9), color(green)(-1), color(purple)(" "8) >B=<9,−1, 8>

=> " " A * B = color(blue)(6 * 9) + color(green)(9 * (-1)) + color(purple)((-1) * 8) = 37⇒ A⋅B=6⋅9+9⋅(−1)+(−1)⋅8=37

||A||||A|| and ||B||||B|| can be computed as follows:

||A|| = sqrt(6^2 + 9^2 + (-1)^2) = sqrt(36 + 81 + 1) = sqrt(118)||A||=√62+92+(−1)2=√36+81+1=√118

||B|| = sqrt(9^2 + (-1)^2 + 8^2) = sqrt(81 + 1 + 64) = sqrt(146)||B||=√92+(−1)2+82=√81+1+64=√146

In total, you have

A * B - ||A|| * ||B|| = 37 - sqrt(118)*sqrt(146) ~~94,26A⋅B−||A||⋅||B||=37−√118⋅√146≈94,26

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