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Question Video: Calculating the Dot Product of Two Dimensional Vectors Mathematics

Given that vector 𝐀 = [−2 and 3] and vector 𝐁 = [−1 and 8], find (𝐀 − 𝐁) ⋅ 𝐁.

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Video Transcript

Given that vector 𝐀 is equal to negative two, three and vector 𝐁 is equal to negative one, eight, find the dot product of 𝐀 minus 𝐁 and 𝐁.

We will begin this question by subtracting vector 𝐁 from vector 𝐀. When subtracting vectors, we simply need to subtract their individual corresponding components. Subtracting negative one from negative two gives us negative one. This is because this is the same as adding one to negative two. Subtracting eight from three gives us negative five. The vector 𝐀 minus 𝐁 is therefore equal to negative one, negative five.

We recall that in order to find the dot or scalar product of two vectors, we begin by finding the product of their individual components and then find the sum of these values. In this question, we need to find the dot product of negative one, negative five and negative one, eight.

Both of the 𝑥-components of these vectors are negative one. The 𝑦-components are negative five and eight. Multiplying two negative numbers gives us a positive answer, whereas multiplying a negative number by a positive number gives a negative answer. This leaves us with one minus 40. Taking 40 away from one gives us an answer of negative 39. This is the dot product of 𝐀 minus 𝐁 and 𝐁.

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