### Video Transcript

Given that five multiplied by vector 𝐀 is equal to negative 10𝐢 minus 15𝐣 plus 10𝐤 and four multiplied by vector 𝐁 is equal to negative eight 𝐢 plus 12𝐣 minus four 𝐤, determine the dot product of four 𝐀 and four 𝐁.

As five multiplied by vector 𝐀 is equal to negative 10𝐢 minus 15𝐣 plus 10𝐤, we can calculate vector 𝐀 by dividing each of the components by five. Vector 𝐀 is therefore equal to negative two 𝐢 minus three 𝐣 plus two 𝐤. Multiplying each of the components of this vector by four, we see that four multiplied by vector 𝐀 is equal to negative eight 𝐢 minus 12𝐣 plus eight 𝐤.

We now have the vectors four 𝐀 and four 𝐁 written in terms of 𝐢, 𝐣, and 𝐤 unit vectors. We recall that if the vectors 𝐮 and 𝐯 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 𝐀 and four 𝐁 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 𝐀 is equal to negative 10𝐢 minus 15𝐣 plus 10𝐤 and four 𝐁 is equal to negative eight 𝐢 plus 12𝐣 minus four 𝐤, then the dot product of four 𝐀 and four 𝐁 is negative 112.