Question Video: Solving Problems Using the Dot Product Mathematics

Given that the vectors 𝐀 = [3 and π‘₯ + 1] and 𝐁 = [βˆ’2π‘₯ and 3] are perpendicular, find the value of π‘₯.

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Video Transcript

Given that the vectors 𝐀 equals three, π‘₯ plus one and 𝐁 equals negative two π‘₯, three are perpendicular, find the value of π‘₯.

The key to this exercise here is to recognize that if 𝐀 and 𝐁 are perpendicular, as they are, then that means that their dot product is equal to zero. This is the case because any vectors that are perpendicular don’t overlap one another at all. The dot product is essentially a measure of overlap, so a dot product of zero between two vectors means that they are perpendicular.

In general, if we have two two-dimensional vectors, we’ll call that 𝐕 one and 𝐕 two, then their dot product is equal to the product of their π‘₯-components added to the product of their 𝑦-components. So when it comes to our given vectors 𝐀 and 𝐁, their dot product equals the π‘₯-value of 𝐀 times the π‘₯-value of 𝐁 plus the 𝑦-value of 𝐀 times the 𝑦-value of 𝐁.

Multiplying out, we have negative six π‘₯ plus three π‘₯ plus three or, simplifying, negative three π‘₯ plus three. We recall that this is equal to zero since 𝐀 and 𝐁 are perpendicular. And so if we subtract three from both sides of this equation, we find that negative three equals negative three π‘₯. And then we can divide both sides by negative three to find that π‘₯ equals one. This is the value of π‘₯ for 𝐀 dot 𝐁 to be zero and, therefore, these vectors to be perpendicular.

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