If #A= <-5 ,3 ,-7 ># and #B= <2 ,2 ,8 >#, what is #A*B -||A|| ||B||#?

#vecA.vecB=<-5,3,-7>.<2,2,8>#

#=-10+6-56=-60#

The magnitude of #vecA# is

#=||vecA||=||<-5,3,-7>||=sqrt((-5)^2+(3)^2+(-7)^2)#

#=sqrt(25+9+49)=sqrt83#

The magnitude of #vecB# is

#=||vecB||=||<2,2,8>||=sqrt((2)^2+(2)^2+(8)^2)#

#=sqrt(4+4+64)=sqrt72#

#vecA.vecB-||vecA||*||vecB||=-60-sqrt83sqrt72#

#=-137.3#

To find \( A \cdot B - \|A\| \|B\| \), where \( A = <-5, 3, -7> \) and \( B = <2, 2, 8> \), we first calculate the dot product of vectors \( A \) and \( B \), then find the magnitudes of vectors \( A \) and \( B \), and finally subtract the product of their magnitudes from the dot product. The dot product of two vectors \( A \) and \( B \) is calculated as: \[ A \cdot B = a_x \cdot b_x + a_y \cdot b_y + a_z \cdot b_z \] Substituting the given values: \[ A \cdot B = (-5 \cdot 2) + (3 \cdot 2) + (-7 \cdot 8) \] \[ A \cdot B = (-10) + (6) + (-56) \] \[ A \cdot B = -10 + 6 - 56 \] \[ A \cdot B = -60 \] The magnitude of a vector \( A \) is given by: \[ \|A\| = \sqrt{a_x^2 + a_y^2 + a_z^2} \] Substituting the given values for vector \( A \): \[ \|A\| = \sqrt{(-5)^2 + 3^2 + (-7)^2} \] \[ \|A\| = \sqrt{25 + 9 + 49} \] \[ \|A\| = \sqrt{83} \] Similarly, the magnitude of vector \( B \) is calculated as: \[ \|B\| = \sqrt{2^2 + 2^2 + 8^2} \] \[ \|B\| = \sqrt{4 + 4 + 64} \] \[ \|B\| = \sqrt{72} \] Now, we can calculate \( A \cdot B - \|A\| \|B\| \): \[ A \cdot B - \|A\| \|B\| = -60 - (\sqrt{83} \cdot \sqrt{72}) \] \[ A \cdot B - \|A\| \|B\| = -60 - (\sqrt{83 \cdot 72}) \] \[ A \cdot B - \|A\| \|B\| = -60 - (\sqrt{5976}) \] \[ A \cdot B - \|A\| \|B\| \approx -60 - 77.36 \] \[ A \cdot B - \|A\| \|B\| \approx -137.36 \] Therefore, \( A \cdot B - \|A\| \|B\| \approx -137.36 \).