Given that the coordinates of 𝐴, 𝐵, and 𝐶, are two and negative four, negative two; negative two, three, three; and four, two, five, respectively, determine the value of the dot product of 𝐴𝐵 and 𝐴𝐶.
We want to find the dot product of the vectors 𝐴𝐵 and 𝐴𝐶. And to do this, we’re either going to need their magnitudes and the measure of the angle between them or we’re going to need their components. We’re not given either set of information directly in the question.
However, we are given the coordinates of points 𝐴, 𝐵, and 𝐶. And from these coordinates, we can work out the components of the vectors 𝐴𝐵 and 𝐴𝐶. Our points have three coordinates and so are in three-dimensional space. And so our vectors are going to be three dimensional as well.
The 𝑥-component of 𝐴𝐵 tells us how far we need to move in the 𝑥-direction from the initial point 𝐴 to the terminal point 𝐵. Looking at the 𝑥-coordinates of 𝐴 and 𝐵, we see that our initial value of 𝑥 is two and our terminal value is negative two. And so we must have moved negative four units.
We do the same calculation for the 𝑦-component, we start at negative four and move to three. So we have moved seven units. To get from negative two to three, our 𝑧-components must be five.
We do the same thing to find the components of the vector 𝐴𝐶. The 𝑥- component is four minus two equals two. The 𝑦-component is two minus negative four, which is six. And the 𝑧-component is five minus negative two, which is seven.
Having found their components, we compute the dot product in the normal way. It’s the product of the 𝑥-components plus the product of the 𝑦-components plus the products of the 𝑧-components. Evaluating this, we find that the dot product is 69.