### Video Transcript

If vector 𝚨 is equal to two 𝐢 minus five 𝐣 minus three 𝐤 and vector 𝚩 is equal to negative four 𝐢 minus two 𝐣 minus 𝐤, find the dot product of vector 𝚨 and vector 𝚩.

We recall that to calculate the dot product of any two vectors, we multiply their corresponding components and then find the sum of these values. In this question, the dot product of vector 𝚨 and vector 𝚩 is equal to two multiplied by negative four plus negative five multiplied by negative two plus negative three multiplied by negative one. Two multiplied by negative four is equal to negative eight. Multiplying two negative numbers gives a positive answer. Therefore, negative five multiplied by negative two is equal to 10. Likewise, negative three multiplied by negative one is equal to three. This leaves us with negative eight plus 10 plus three. Negative eight plus 10 is equal to two, and adding three to this gives us five.

If vector 𝚨 is equal to two 𝐢 minus five 𝐣 minus three 𝐤 and vector 𝚩 is equal to negative four 𝐢 minus two 𝐣 minus 𝐤, then the dot product of vector 𝚨 and vector 𝚩 is equal to five.