Video Transcript
Given that five multiplied by vector ๐ is equal to negative 10๐ข minus 15๐ฃ plus 10๐ค and four multiplied by vector ๐ is equal to negative eight ๐ข plus 12๐ฃ minus four ๐ค, determine the dot product of four ๐ and four ๐.
As five multiplied by vector ๐ is equal to negative 10๐ข minus 15๐ฃ plus 10๐ค, we can calculate vector ๐ by dividing each of the components by five. Vector ๐ is therefore equal to negative two ๐ข minus three ๐ฃ plus two ๐ค. Multiplying each of the components of this vector by four, we see that four multiplied by vector ๐ is equal to negative eight ๐ข minus 12๐ฃ plus eight ๐ค.
We now have the vectors four ๐ and four ๐ written in terms of ๐ข, ๐ฃ, and ๐ค unit vectors. We recall that if the vectors ๐ฎ and ๐ฏ are written as shown, we can calculate their dot product. We do this by multiplying the corresponding components and then finding the sum of these three values. The dot product of four ๐ and four ๐ is therefore equal to negative eight multiplied by negative eight plus negative 12 multiplied by 12 plus eight multiplied by negative four. Negative eight multiplied by negative eight is 64. Negative 12 multiplied by 12 is equal to negative 144. And eight multiplied by negative four is negative 32. The sum of these three values is negative 112.
If five ๐ is equal to negative 10๐ข minus 15๐ฃ plus 10๐ค and four ๐ is equal to negative eight ๐ข plus 12๐ฃ minus four ๐ค, then the dot product of four ๐ and four ๐ is negative 112.
