Video: Finding the Unknown Value of Two Perpendicular Vectors

Given that 𝐀 = <2, 2, −6>, 𝐁 = <3, 3, 𝑘>, and 𝐀 and 𝐁 are two perpendicular vectors, find the value of 𝑘.

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Video Transcript

Given that 𝐀 is equal to two, two, negative six; 𝐁 is equal to three, three, 𝑘; and 𝐀 and 𝐁 are two perpendicular vectors, find the value of 𝑘.

We begin by recalling that two vectors are perpendicular if their dot or scalar product is equal to zero. This means that we begin this question by finding the dot product of vector 𝐀 and vector 𝐁, where vector 𝐀 is equal to two, two, negative six and vector 𝐁 is equal to three, three, 𝑘.

We calculate the dot product by multiplying the corresponding components and then finding the sum of these values. In this question, we have two multiplied by three plus two multiplied by three plus negative six multiplied by 𝑘. This simplifies to six plus six plus negative six 𝑘, which in turn is equal to six plus six minus six 𝑘. Six plus six is equal to 12. And as these vectors are perpendicular, we know that 12 minus six 𝑘 is equal to zero. Adding six 𝑘 to both sides of this equation, we have six 𝑘 is equal to 12. We can then divide both sides of this equation by six, giving us 𝑘 is equal to two.

If the vectors two, two, negative six and three, three, 𝑘 are perpendicular, then 𝑘 is equal to two.

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