Given that 𝐀 is equal to five, three, negative one; 𝐁 is equal to two, two, 𝑘; and 𝐀 and 𝐁 are two perpendicular vectors, find the magnitude of vector 𝐁.
We begin by recalling that two vectors 𝐮 and 𝐯 are perpendicular if their dot or scalar product is equal to zero. In this question, we begin by finding the dot product of vectors 𝐀 and 𝐁 in terms of 𝑘. We find the dot product of any two vectors by multiplying their corresponding components and then finding the sum of these values.
In this question, we have five multiplied by two plus three multiplied by two plus negative one multiplied by 𝑘. As the vectors are perpendicular, we know that this is equal to zero. Five multiplied by two is equal to 10, three multiplied by two is equal to six, and negative one multiplied by 𝑘 is negative 𝑘. 10 plus six plus negative 𝑘 is equal to zero. This simplifies to zero is equal to 16 minus 𝑘. By adding 𝑘 to both sides of this equation, we have a value of 𝑘 equal to 16. This means that vector 𝐁 is equal to two, two, 16.
We now need to calculate the magnitude of this vector. The magnitude of any vector is equal to the square root of the sum of the squares of its individual components. The magnitude of vector 𝐁 is therefore equal to the square root of two squared plus two squared plus 16 squared. Two squared is equal to four, and 16 squared is 256. The magnitude of vector 𝐁 is therefore equal to the square root of 264. Using our laws of radicals or surds and the fact that four multiplied by 66 is 264, root 264 is equal to root four multiplied by root 66. As the square root of four is equal to two, the magnitude of vector 𝐁, which must be positive, is equal to two root 66.