### Video Transcript

On a lattice where 𝐀𝐂 is equal to negative five, negative five; 𝐁𝐂 is equal to negative 12, six; and three multiplied by vector 𝐂 plus 𝐀𝐁 is equal to negative eight, 13, determine the coordinates of the point 𝐁.

We know that the vector 𝐀𝐁 is equal to the vector 𝐀𝐂 plus the vector 𝐂𝐁. As the vector 𝐁𝐂 has the same magnitude but opposite direction of the vector 𝐂𝐁, then 𝐀𝐁 is equal to 𝐀𝐂 minus 𝐁𝐂. The vector 𝐀𝐁 is, therefore, equal to negative five, negative five minus negative 12, six. We can subtract two vectors by subtracting their corresponding components. Negative five minus negative 12 is the same as negative five plus 12, which equals seven. Negative five minus six is equal to negative 11. Therefore, the vector 𝐀𝐁 is equal to seven, negative 11.

This means that three multiplied by vector 𝐂 plus seven, negative 11 is equal to negative eight, 13. We can subtract the vector seven, negative 11 from both sides. Three multiplied by vector 𝐂 is, therefore, equal to negative 15, 24. We can then divide both sides of this equation by three such that vector 𝐂 is equal to negative five, eight.

We can then use the fact that the vector 𝐁𝐂 is equal to the vector 𝐂 minus the vector 𝐁. Rearranging this, the vector 𝐁 is equal to the vector 𝐂 minus the vector 𝐁𝐂. Substituting in the vectors we know, vector 𝐁 is equal to negative five, eight minus negative 12, six. Once again, we subtract the corresponding components such that vector 𝐁 is equal to seven, two.

As the vector 𝐁 is its displacement from the origin, the coordinates of the point 𝐁 are seven, two.