# Question Video: Determining the Dot Product between Vectors Mathematics

Given that the coordinates of π΄, π΅, and πΆ are (2, β4, β2), (β2, 3, 3), and (4, 2, 5), respectively, determine π΄π΅ β π΄πΆ.

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### Video Transcript

Given that the coordinates of π΄, π΅, and πΆ, are two and negative four, negative two; negative two, three, three; and four, two, five, respectively, determine the value of the dot product of π΄π΅ and π΄πΆ.

We want to find the dot product of the vectors π΄π΅ and π΄πΆ. And to do this, weβre either going to need their magnitudes and the measure of the angle between them or weβre going to need their components. Weβre not given either set of information directly in the question.

However, we are given the coordinates of points π΄, π΅, and πΆ. And from these coordinates, we can work out the components of the vectors π΄π΅ and π΄πΆ. 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 π₯-component of π΄π΅ tells us how far we need to move in the π₯-direction from the initial point π΄ to the terminal point π΅. Looking at the π₯-coordinates of π΄ and π΅, we see that our initial value of π₯ 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 π¦-component, we start at negative four and move to three. So we have moved seven units. To get from negative two to three, our π§-components must be five.

We do the same thing to find the components of the vector π΄πΆ. The π₯- component is four minus two equals two. The π¦-component is two minus negative four, which is six. And the π§-component is five minus negative two, which is seven.

Having found their components, we compute the dot product in the normal way. Itβs the product of the π₯-components plus the product of the π¦-components plus the products of the π§-components. Evaluating this, we find that the dot product is 69.