Given that vector 𝐀 is equal to three, four, four; vector 𝐁 is equal to four, three, 𝑘; and 𝐀 and 𝐁 are two perpendicular vectors, find the value of 𝐀 minus 𝐁.
We begin by recalling that if two vectors 𝐮 and 𝐯 are perpendicular, their dot or scalar product is equal to zero. This means that we will begin this question by finding the dot product of vector 𝐀 and vector 𝐁 in terms of 𝑘. To find the dot product of any two vectors, we multiply the corresponding components and then find the sum of these values.
In this question, we have three multiplied by four plus four multiplied by three plus four multiplied by 𝑘. And as the vectors are perpendicular, we know this is equal to zero. The right-hand side simplifies to 12 plus 12 plus four 𝑘. As 12 plus 12 is 24, we have 24 plus four 𝑘 is equal to zero. We can then subtract 24 from both sides of this equation, giving us four 𝑘 is equal to negative 24. Dividing both sides of this equation by four gives us 𝑘 is equal to negative six. This means that vector 𝐁 is equal to four, three, negative six.
We need to subtract this vector from vector 𝐀. This gives us three, four, four minus four, three, negative six. We then subtract the corresponding components. Three minus four is equal to negative one, four minus three is equal to one, and four minus negative six is equal to 10. If 𝐀 and 𝐁 are two perpendicular vectors, then 𝐀 minus 𝐁 is equal to negative one, one, 10.